Scaling Limits of Random Planar Structures

103 downloads 0 Views 1MB Size Report
My supervisor, Louigi Addario-Berry, is a coauthor of the paper Joint Conver- ...... construction is analogous to the Brownian map but we use R rather than Brow-.
Critical and Subcritical Scaling Limits of Random Planar Maps with Connectivity Constraints

by

Yuting Wen The Department of Mathematics and Statistics ´al McGill University, Montre March 2016

A thesis submitted to McGill University in partial fulfilment of the requirements for the degree of

Doctor of Philosophy in Combinatorial Probability c

Yuting Wen 2016

Contents 0.1. 0.2. 0.3. 0.4.

Abstract R´esum´e Acknowledgements Preface and Contribution of Authors

iv vi viii x

List of notation and terminology

xi

Chapter 1. Introduction 1.1. The Brownian Map as a Scaling Limit of Random Maps 1.2. The Brownian Plane as a Non-Compact Limit of Random Maps 1.3. Scaling Limits of Maps with Higher Connectivity 1.4. Notation

2 3 15 20 23

Chapter 2. Several Concepts about Maps 2.1. Graphs and Maps 2.2. Labelled Trees and the BDG Bijection 2.3. Map Enumeration and Singularity Analysis

25 25 28 34

Chapter 3. Joint Convergence of Random Quadrangulations and Their Cores 38 3.1. Introduction 38 3.2. Preliminaries 51 3.3. Composition Schemata 52 3.4. Airy Distribution for Quadrangulations 57 3.5. Sizes and Diameters of Pendant Submaps 63 3.6. Exchangeable Decorations 70 3.7. Projection of Masses in Random Quadrangulations 74 3.8. Proofs of the Main Theorems 80 3.9. Tutte’s Bijection and the Quadratic Method 87 3.10. Remaining Derivation Using Singularity Analysis 89 Chapter 4. The Brownian Plane with Minimal Neck Baby Universe 4.1. Introduction 4.2. Pointed and Local Gromov-Hausdorff-Prokhorov Distances 4.3. Map Decomposition and Balls-in-Boxes Model 4.4. Asymptotically Stable Distribution -ii-

93 93 98 99 103

4.5. Random Allocation with Varying Balls-to-Boxes Ratio 106 4.6. The Number of Facial 2-Cycles in the Pre-Root-Block 114 4.7. Condensation in Uniform Quadrangulation Conditioned on Root Block Size 116 4.8. Uniformly Asymptotically Negligible Attachments 120 4.9. Proofs of the Main Theorems 127 4.10. The Brownian Plane with Minbus 130 4.11. Convergence to the Brownian Plane in the Gromov-HausdorffProkhorov Topology 132 4.12. Expected Size of Pendant Submap 138 Chapter 5. Conclusions 5.1. Summary of Main Results 5.2. Summary of Methodologies

141 141 142

Chapter 6. Potential Future Works 6.1. Concentration of the Total Mass of Non-Largest Pendant Submaps 6.2. The Scaling Limit of 3-Connected General Maps 6.3. Properties of Large Planar Graphs

144 144 151 151

Bibliography

154

-iii-

0.1. Abstract In the first part, we show that a uniform quadrangulation, its largest 2-connected block, and its largest simple block, upon rescaling the graph distance properly, jointly converge to the same Brownian map in distribution for the Gromov-HausdorffProkhorov topology. We start by deriving a local limit theorem for the asymptotics of maximal block sizes, extending the result by Banderier, Flajolet, Schaeffer & Soria [16]. The resulting diameter bounds for pendant submaps of random quadrangulations straightforwardly lead to Gromov-Hausdorff convergence. To extend the convergence to the Gromov-Hausdorff-Prokhorov topology, we show that exchangeable “uniformly asymptotically negligible” attachments of mass simply yield, in the limit, a deterministic scaling of the mass measure. In the second part, for each n P N, let Qn be a uniform rooted measured quadrangulation of size n conditioned to have rpnq vertices in its root block. We prove ´ ¯1{4 21 that for a suitable function rpnq, after rescaling graph distance by 40¨rpnq , with an appropriate rescaling of measure, Qn converges to a random pointed measured non-compact metric space S, in the local Gromov-Hausdorff-Prokhorov topology; the space S is built by identifying a uniform point of the Brownian map with the distinguished point of the Brownian plane. Our result relies upon both the convergence of uniform quadrangulations towards the Brownian plane by Curien & Le Gall [30], and the convergence of uniform 2-connected quadrangulations to the Brownian map, proved in the first part of the thesis. The main steps of the proof are as follows. First, we show that the sizes of submaps pendant to the root block have an asymptotically stable distribution. Second, we deduce asymptotics for occupancy in a random allocation model with a varying balls-to-boxes ratio. Third, we establish a bound for the number of pendant submaps of the root block, which allows us to apply the occupancy bounds to uniformly control the sizes of pendant submaps.

-iv-

This entails that the pendant submaps act as uniformly asymptotically negligible “decorations” which do not affect the scaling limit.

Keywords: scaling limit, Brownian map, Brownian plane, Gromov-HausdorffProkhorov topology, singularity analysis, quadrangulation, connectivity, core, condensation, random allocation.

-v-

0.2. R´ esum´ e Dans la premi`ere partie, nous d´emontrons qu’une quadrangulation uniforme, son plus grand bloc 2-connexe, et son plus grand bloc simple, convergent conjointement, apr`es rescaling, vers la carte brownienne, en distribution pour la topologie de Gromov-Hausdorff-Prokhorov. Nous commen¸cons par obtenir un th´eor`eme limite locale pour les asymptotique des tailles de blocs maximales, ´etendant le r´esultat par Banderier, Flajolet, Schaeffer & Soria [16]. En r´esultent des bornes sur le diam`etre des sous-cartes pendantes d’une quadrangulation al´eatoire, qui permettent de garantir la convergence au sens de Gromov-Hausdorff. Pour obtenir la convergence au sens de la topologie de Gromov-Hausdorff-Prokhorov, nous montrons que des ajouts de masse ´echangeables et “uniform´ement et asymptotiquement n´egligeables” donnent tout simplement `a la limite la mˆeme mesure de masse, a` changement d’´echelle d´eterministe pr`es. Dans la deuxi`eme partie, pour chaque n P N, soit Qn une quadrangulation uniforme de taille n, enracin´ee, mesur´ee et conditionn´ee `a avoir rpnq sommets dans son bloc racine. Nous prouvons que pour une fonction appropri´ee rpnq, apr`es multiplica´ ¯1{4 21 tion de la distance de graphe par 40¨rpnq , la quadrangulation Qn converge dans la topologie locale de Gromov-Hausdorff-Prokhorov vers S un espace m´etrique al´eatoire non-compact, point´e et mesur´e; l’espace S est construit par identification d’un point uniforme de la carte brownienne avec le point distingu´e du plan brownien. Notre r´esultat repose `a la fois sur la convergence des quadrangulations uniformes vers le plan brownien par Curien & Le Gall [30], et la convergence des quadrangulations uniformes 2-connexes vers la carte brownienne, r´ecemment prouv´ee par Addario-Berry & Wen [5]. Les principales ´etapes de la preuve sont les suivantes. Premi`erement, nous d´emontrons que les tailles des sous-cartes pendantes au bloc racine suivent asymptotiquement une loi stable. Deuxi`emement, nous prouvouns des asymptotiques pour les taux d’occupation dans un mod`ele d’allocation al´eatoire avec un ratio “boules-aux-boˆıtes” variant. Troisi`emement, nous ´etablissons une borne pour le nombre de sous-cartes pendantes du bloc racine, ce qui nous permet d’appliquer -vi-

les r´esultats du mod`ele d’allocation pour contrˆoler uniform´ement les tailles des souscartes pendantes. Cela implique que les sous-cartes pendantes agissent comme des “d´ecorations” uniform´ement asymptotiquement n´egligeables, qui n’affectent pas la limite d’´echelle.

Mots-clefs: limite d’´echelle, carte brownienne, plan brownien, topologie de Gromov-Hausdorff-Prokhorov, analyse des singularit´es, quadrangulations, connectivit´e, noyau, condensation, allocation al´eatoire.

-vii-

0.3. Acknowledgements First and foremost, I would like to express deepest gratitude to my supervisor Louigi Addario-Berry. During these 3 years, Louigi keeps inspiring me with his unique spirits and insights into mathematics, generously giving me countless hours and valuable advice when needed. I truly appreciate the opportunity I have had to learn and grow as a mathematician under his guidance. I am thankful to Professors Guillaume Chapuy and Luc Devroye for careful reviews on the initial draft of this thesis and for helpful suggestions on the open problems. I also thank Emmanuel Jacob for proofreading the French abstract. In addition, I would like to thank Professor Gesine Reinert for her encouragement and stimulating discussions during my visit at the University of Oxford. Her generous guidance has deepened my interest in coupling methods and random networks. I am also very thankful to Professor Christina Goldschmidt for organizing my visit to the University of Oxford which I immensely enjoyed, and to numerous peers, professors, and staffs for their hospitality while I was at Oxford. Furthermore, I thank Marie Albenque for patiently explaining her work and ideas about bijections between blossoming trees and maps. I also thank Nicolas Broutin for the fruitful discussions about spanning triangulations and branching processes, and for arranging my visit to the INRIA Institute in Paris-Rocquencourt. My thanks also go to all the researchers in the RAP team at INRIA for their hospitality during my visit. I am grateful to many professors at McGill University from whom I have learnt so much, including Bruce Reed for carefully grading my assignments in structural graph theory, Luc Devroye for his enthusiastic lectures in probabilistic methods, Linan Chen for her very well-prepared and enriching lectures in probability, Johanna Neˇslehov´a for her fascinating lectures in extreme value theory, Hamed Hatami for the useful discussions about graph limits and interesting lectures in additive combinatorics, and Sergey Norin, David Stephens, Chritian Genest for their wonderful lectures, et cetera. -viii-

My time at McGill University has gifted me with so many friends, and it would not have been nearly as much fun without the friendship of Laura Eslava, Tao Lei, Liana Yepremyan, Jonathan Noel, Lena Yuditsky, Xing Shi Cai, Guillem Perarnau Llobet, Mashbat Suzuki, Richard Santiago, Jinming Wen, Siyuan Lu, Xianchao Wu, Orla Murphy, Meng Zhao, Huijun Chen, and many other wonderful people. There are people who I could never thank enough, especially my parents who love me unconditionally and support me if needed. Finally, I thank Professor Louigi Addario-Berry, Professor Johanna Neˇslehov´a, the Department of Mathematics and Statistics at McGill University, Montr´eal’s Centre de Recherches Math´ematiques, and Fonds Qubcois de la Recherche sur la Nature et les Technologies for their significant financial support during my PhD study. I also thank the staffs at the Department of Mathematics and Statistics at McGill University for their helps from time to time.

-ix-

0.4. Preface and Contribution of Authors My supervisor, Louigi Addario-Berry, is a coauthor of the paper Joint Convergence of Random Quadrangulations and Their Cores [5]. This paper constitutes most of the contents of Chapter 3. The ideas in this chapter were developed jointly. The text was written primarily by myself, with the exception of Sections 3.6 and 3.7. Chapter 4 is based on the paper The Brownian Plane with Minimal Neck Baby Universe [68], authored by myself. The contents of Chapters 3, 4, 5, and 6 of this thesis are original scholarship and distinct contributions to knowledge.

-x-

List of notation and terminology A the Airy density function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 Composition schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Singular expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Singular with exponent 3{2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Map schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 the BDG bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 Boltzmann distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The CRT ....................................................................8 ISE ....................................................................5 m8 the Brownian map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 P the Brownian plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 The Brownian snake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 dGH the Gromov-Hausdorff distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 dGHP the Gromov-Hausdorff-Prokhorov distance . . . . . . . . . . . . . . . . . . . . . . . . 50 dH the Hausdorff distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 dLGHP the local Gromov-Hausdorff-Prokhorov distance . . . . . . . . . . . . . . . . . . . 96 dP the L´evy-Prokorov distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ‹ dGHP the pointed Gromov-Hausdroff-Prokhorov distance . . . . . . . . . . . . . . . . 96 2-connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 dG graph distance on graph G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 c¨d pc ¨ dqpx, yq “ c ¨ dpx, yq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 epGq the edge set of graph G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 GrV s subraph of G induced by the vertex set V . . . . . . . . . . . . . . . . . . . . . . . . . 24 Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 G´V GrvpGqzV s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Multiple edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Simple graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Graph size ....................................................................3 degG pvq the degree of vertex v in graph G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 vpGq the vertex set of graph G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 intpeq interior of the curve e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2-connected block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 RpMq the largest 2-connected block of the rooted map M (Chapter 3). . . .39

-xi-

bpMq R‚ pMq SpMq sbpMq S‚ pMq Simple block DpRr q cG µG Nearly facial Nearly simple Q R S Qq,r,s Rr,s Map Corner Embedded graph Face degree Bm,k RpQq R` pQq FpQq kn ΛpQq νG µnG pV q ν Pi pQq Pre-root-block ppkq ρQ ξ ăM ăM q-angulation d Ñ p Ñ N

the size of the largest 2-connected block of the rooted map M . . . . . 39 the 2-connected root block of the rooted map M (Chapter 3) . . . . . . 39 the largest simple block of the rooted map M . . . . . . . . . . . . . . . . . . . . . 39 the size of the largest simple block of the rooted map M . . . . . . . . . . . 39 the simple root block of the rooted map M . . . . . . . . . . . . . . . . . . . . . . . . 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 max pdiampΘi q : 0 ď i ď 2sprq ´ 4q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 pvpGq, c ¨ dG , µG q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 the uniform probability over vpGq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 the set of rooted quadrangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 the set of rooted 2-connected quadrangulations . . . . . . . . . . . . . . . . . . . . 37 the set of rooted simple quadrangulations . . . . . . . . . . . . . . . . . . . . . . . . . 37 tQ P Qq : bpQq “ r, sbpQq “ su . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 tQ P Rr : sbpQq “ su . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ....................................................................3 allocation of m balls in k boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 the root block of the rooted quadrangulation Q (Chapter 4) . . . . . . . 91 Q ´ vpLpQqqztρQ u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 the set of facial 2-cycles in the pre-root-block of Q . . . . . . . . . . . . . . . . 91 ´ ¯1{4 40¨rpnq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 21 the largest submap pendant to the root block of the rooted quadrangulation Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 ř δv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 řvPvpGq ti:vi PV u ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 E rξs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 submap pendant to the root block of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 a 3{2-stable distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 vpRpQqq X vpΛpQqq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 a random variable with distribution ppkq . . . . . . . . . . . . . . . . . . . . . . . . . 100 a total order on epMq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 a total order on vpMq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 convergence in distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 convergence in probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 t1, 2, . . .u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

-xii-

Ně0 t0, 1, . . .u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 G Pu G G is chosen uniformly at random from G . . . . . . . . . . . . . . . . . . . . . . . . . . 22 With high probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 kT pwq the number of children of vertex w in the planted plane tree T . . . . 28 Cθ contour process of the p-tree θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 rT contour exploration of the planted plane tree T . . . . . . . . . . . . . . . . . . . 27 kT pw, iq the i-th child of vertex w in the planted plane tree T . . . . . . . . . . . . . . 28 Zθ label process of the p-tree θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Plane tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Planted plane tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 p-tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 BT the set of black vertices of the p-tree T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 pT, uv, Xq labelled p-tree with valid labelling X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 XT valid labelling of the p-tree T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 XT pvq label of v in the p-tree T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 WT the set of white vertices of the p-tree T. . . . . . . . . . . . . . . . . . . . . . . . . . . .28

-xiii-

Introduction

2

CHAPTER 1

Introduction This chapter aims to provide a terse introduction to the progress of the research on scaling limits of random maps and related topics in the past two decades. We focus on presenting our main contributions and situating them within the literature on random maps. We refer the reader to Chapter 2 and the references therein for more rigorous definitions of maps. Section 1.1 gives an introduction to the Brownian map which is the scaling limit of various families of maps. Section 1.2 explains that the Brownian plane is a noncompact scaling limit which possesses the scaling invariance property. In Section 1.3, we describe the main results on which this thesis is based. More specifically, we discuss maps with connectivity constraints, namely the results on simple triangulations and quadrangutions by Addario-Berry & Albenque [4], and the results on the critical (resp. subcritical) scaling limit of quadrangulations with connectivity constraints by Addario-Berry & Wen [5] (resp. Wen [68]). The works of [5] and [68] constitute the majority of this thesis. Finally, Section 1.4 introduces a few notations that we use throughout the thesis. We begin with a few notions on maps. A (multi)graph is connected if there is a path between any two vertices. A (planar) map is a finite connected graph drawn on the the 2-sphere without edge-crossing and it is considered up to orientationpreserving homeomorphisms of the sphere. All graphs considered in this thesis are planar and connected, unless stated otherwise. Faces of a map are the connected components of the complement of edges. The degree of a face equals the number of edges incident to the face; if both sides of the edge are incident to the face then this edge is counted twice. In most part of the thesis we consider quadrangulations (all faces have degree four), sometimes triangulations (all faces have degree three),

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

q-angulations (all faces have degree q) for q P N with q ě 3, bipartite maps (all faces have even degree), and occasionally other families of maps. It is necessary in most cases to consider rooted maps, which are pairs pM, uvq where M is a map and uv is an oriented edge of M . Sometimes it is convenient to consider pointed and rooted maps, which are triples pM, uv, oq where pM, uvq is a rooted map and o is a vertex of M . Given a map M , write vpM q, epM q, and f pM q, respectively, for the vertex set, the edge set, and the set of faces of M . For a set V, write |V| for the cardinality of V. The size of a map M is the number of vertices, |vpM q|. Given a map M , by Euler’s formula, we have |vpM q| ` |f pM q| ´ |epM q| “ 2 . For a quadrangulation Q, it is easily seen that |epQq| “ 2|f pQq| , so |vpQq| “ |f pQq| ` 2 . It is handy to keep this in mind throughout. 1.1. The Brownian Map as a Scaling Limit of Random Maps How does a “uniformly random” metric on the 2-sphere S2 look? Rooted in the theory of two-dimensional quantum gravity, this question has attracted the attention of numerous physicists and mathematicians in the past two decades, partially because studying it helps us to generalize Feynman path integrals to Feynman integrals over surfaces. See Ambj∅rn, Durhuus & J´onsson [12] for an introduction on quantum geometry. One approach to understanding a uniformly random metric on the 2-sphere is to randomly discretize the 2-sphere into a triangulation or other map, then try to find a continuous limit as the size of the map tends to infinity upon proper rescaling of the graph distance. It is natural to conjecture that a Brownian-type surface should arise as a limit of discretizations of the 2-sphere, just

3

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

as Brownian motion is a universal limit for discrete random paths satisfying mild integrability conditions. It took much effort to establish the existence of such a continuous limit. In the 2000s, mathematicians and physicists found out that several families of maps share the same asymptotic diameter and other statistics, and proved a convergence towards a limit object, called the Brownian map; see [49, 50] for a survey. However, the first topology in which that convergence was proved does not contain enough information to yield convergence of many natural functionals; see [49]. Then Le Gall [46] established the convergence along a subsequence in a stronger topology, but it took a few more years until the uniqueness of the Brownian map was independently proven by Miermont [59] and Le Gall [48]. We review the works prior to [46] in Section 1.1.1, followed by an overview on the main results of [48, 59] in Section 1.1.2. Then in Section 1.1.3, we present the construction of the Brownian map in terms of a Gaussian process indexed by the Brownian Continuum Random Tree (CRT), as appeared in [46, 48]. See Section 1.1.3 for these definitions of CRT, and refer to Aldous’ papers [8, 9] for greater details. In Section 1.1.4, we briefly overview the results of convergence to and uniqueness of the Brownian map. Finally, we give a brief account of recent progress on proving universality of the Brownian map in Section 1.1.5. Interested readers are referred to the papers by Le Gall & Miermont [50] and Le Gall [49] for omitted surveys.

1.1.1. Diameters and Profiles of Large Random Maps. In the remaining chapter we assume familiarity with Brownian excursion. The Brownian snake is a strong Markov process taking values in the space of all stopped paths. See Section 1.1.3 for the definition of Brownian snake, and we refer the reader to [44] by Le Gall for a complete exposition. The Integrated SuperBrownian Excursion (ISE) is the random occupation measure of the Brownian snake with the normalized Brownian excursion lifetime process. It was introduced by Aldous [10] as a model of random distributions of masses, briefly reviewed as follows.

4

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

Fix n P N. Let T be an abstract rooted tree with n vertices, say a Cayley tree, taken from the uniform distribution. We embed T into the lattice on Zd for some appropriate d P N, with the root of T at the origin and edges randomly mapped on edges of the lattice. Assigning masses to leaves of T then yields a random distribution of mass on Zd . The random distribution of mass converges, upon scaling the lattice by n´1{4 , to a random measure on Rd , called the ISE. The pioneering work by Chassaing & Schaeffer [28] found a connection between uniform rooted quadrangulations and ISE. More precisely, for n P N, write rn for the radius of a uniform rooted quadrangulation with n faces, and let r be the support of the one-dimensional ISE, then ´1{4

n

ˆ ˙1{4 8 rn Ñ r 9

in distribution as n Ñ 8; see [28, Corollary 3]. Furthermore, the scaled profile (the number of vertices at a given distance from the distinguished vertex) converges in distribution to the mass of the ISE. Their work suggested the existence of a continuum random map as a limit, scaled by n´1{4 , of uniform rooted quadrangulations. Subsequent to the work of [28], Marckert & Mokkadem [53] confirmed the existence of such continuous and compact limit of uniform pointed quadrangulations, and coined the name Brownian map, which is defined using the Brownian snake. (A pointed quadrangulation is a quadrangulation paired with a distinguished vertex.) The same result also holds for uniform rooted quadrangulations, and some rooted quadrangulations with random edge lengths. Their proof relies on identifying a quadrangulation with the gluing of two trees using Schaeffer’s bijection [28, 62], then showing that the pair of trees converges weakly to a pair of random continuous trees. The convergence holds in a metric space of rooted abstract maps defined in [53], which contains rooted quadrangulations and the Brownian map. Yet, the topology of this metric space is not as nice as the Gromov-Hausdorff (GH) topology. See Section 3.2.1 for the definition of the GH topology.

5

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

On the other hand, Le Gall [45] proved an invariance principle for discrete labelled trees [51] with the condition that the labels stay positive. From the invariance principle he deduces that, the tail of the root edge in a uniform rooted quadrangulation Qn with n faces has approximately the same law of a uniform vertex in Qn as n Ñ 8. By avoiding the use of Vervaat transformation to define conditionally positive tree-indexed Gaussian process, Le Gall recovered and strengthened the results of [28]. In [25], Bouttier, Di Francesco & Guitter invented a bijection which generalizes Schaeffer’s bijection to more general families of maps. We describe this bijection, which we hereafter call the BDG bijection, in Section 2.2.3. Using the BDG bijection, Marckert & Miermont [52] established invariance principles for uniform pointed and rooted bipartite maps under suitable Boltzmann distributions (the weight of a ś bipartite map M is f Pf pM q wdegpf q where pw2i : i P Ně0 q is a sequence of nonnegative real numbers, and degpf q denotes the degree of f in M ; the corresponding ś Boltzmann distributed random map M has P pM “ M q 9 f Pf pM q wdegpf q ). Weill [67] proved the same results as [52] for uniform rooted bipartite maps under Boltzmann distribution. That work investigates distances from the root, instead of from the distinguished point as in [52], and it has an application in Le Gall’s paper [46]. Miermont [56] generalized the result of [52] to uniform pointed and rooted Boltzmann maps without constraint on face degrees, by using the BDG bijection as well. In this case, the BDG bijection is more delicate to use than the one encoding bipartite maps. Thus, a more powerful invariance principle for multitype trees is needed, which was proved by Miermont [57]. The invariance principle of [57] later became an ingredient for the proof of the milestone theorem, reviewed in Section 1.1.2. Later, Miermont & Weill [60] proved the same results as [56] in the rooted case. At around the same time, in a collection of open problems on conformally invariant scaling limits by Schramm [63], a precise formulation was given for the scaling limit of maps in the GH topology. Now we briefly describe such a formulation. Given a graph G, write dG for the graph distance. Given a metric space pV, dq, for c ě 0,

6

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

we write c ¨ d for the metric on V such that pc ¨ dqpx, yq “ c ¨ dpx, yq . For each n P N, let Mn be a uniformly chosen element from a family of maps with n ˆ n “ pvpMn q, n´1{4 ¨ dMn q. Are there random metric spaces M and Z faces, and let M ˆ n and M can be isometrically embedded in Z, in a way that the Hausdorff such that M ˆ n and M tends to 0 with n (in distribution or distance between the embeddings of M almost surely)? See Section 3.2.1 for the definition of Hausdorff distance. The aforementioned papers on radius and profiles of large random maps did not establish the convergence to the Brownian map for the GH topology. Nonetheless, their results on scaling limits of labelled trees, and on radius and profiles of random maps, allow ones to control distances from a distinguished vertex. One key missing ingredient for establishing GH convergence is the joint distribution of distances (a matrix) between an arbitrary fixed number k of vertices. However, the above results paved the way for the GH convergence result to come.

1.1.2. Milestone Theorems. Le Gall [46] proved the existence of subsequential Gromov-Hausdorff (GH) scaling limits for uniform rooted 2p-angulations with n faces, for integer p ě 2. He uses the term “Brownian map” for any random compact metric space arising as such a limit. The paper [46] uses a compactness argument which requires the extraction of proper subsequences. The uniqueness of the Brownian map had not been proven by that time, due to the difficulty of characterizing Brownian map completely – only upper and lower bounds for the distance between two uniform points of the Brownian map were available. Subsequently, Le Gall [47] showed that there is an almost surely unique geodesic path between a uniform point in M and the projection of a non-leaf random point in the Brownian CRT. Building on top of the previous two results, Le Gall [48] proved the uniqueness of the Brownian map.

7

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

Theorem 1.1.1 ([48, Theorem 1.1]) Set ˆ cq “

9 qpq ´ 2q

˙1{4

for even integer q ě 2, and set cq “ 61{4 if q “ 3. Let Mn be a uniform element from the set of pointed and rooted q-angulations with n facesa. There exists a random compact metric space pM, D1˚ q called the Brownian map, which does not depend on q, such that `

˘ vpMn q, cq n´1{4 ¨ dMn Ñ pM, D1˚ q

as n Ñ 8 in distribution for the Gromov-Hausdorff topology. aEquivalently,

we may sample Mn uniformly from the set of rooted q-angulations with n faces then couple it with a uniform vertex. Similarly for Qn in the next theorem.

Just over a month before (we state Le Gall’s result first as this makes the presentation more coherent), Miermont [59] posted an independent proof of the above theorem in the case of quadrangulations, i.e., for q “ 4. Theorem 1.1.2 ([59, Theorem 1]) Let Qn be a uniform pointed and rooted quadrangulations with n faces. There exists a random compact metric space pM, D1˚ q called the Brownian map such that

˜

ˆ vpQn q,

9 8n

¸

˙1{4 ¨ dQn

Ñ pM, D1˚ q

as n Ñ 8 in distribution for the Gromov-Hausdorff topology.

These theorems are milestones in finding a uniform random metric on the 2sphere and showing ramifications in some other branches of mathematics such as graph enumeration. They are also foundational for the work of the current thesis.

8

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

9

1.1.3. Le Gall’s Construction of the Brownian Map. For s, t P R, we write s ^ t “ minps, tq and s _ t “ maxps, tq. Let e be a standard Brownian excursion. The Brownian Continuum Random Tree is the quotient space Tp1q :“ r0, 1s{ „e , where s „e t if es “ et “

inf rPrs^t,s_ts

er ,

for s, t P r0, 1s. We let pp1q : r0, 1s Ñ Tp1q be the canonical projection, sending each x P r0, 1s to its equivalence class in Tp1q . Next, define a process Z “ pZt qtPr0,1s such that, conditional on e, Z is a centred Gaussian process with covariance E rZs Zt | es “

inf rPrs^t,s_ts

er

for s, t P r0, 1s. We may and shall view Z as an almost surely continuous sample path [17] indexed by Tp1q , and call it the head of the Brownian snake. Furthermore, for s, t P r0, 1s with s ď t, let ˆ D1˝ ps, tq



D1˝ pt, sq

“ Zs ` Zt ´ 2 max

˙ min Zr ,

rPrs,ts

min rPr0,ssYrt,1s

Zr

.

Clearly, the function D1˝ is a pseudo-metric on r0, 1s2 . Then, for a, b P Tp1q , define ` ˘ D1˝ pa, bq “ inf D1˝ ps, tq : s, t P r0, 1s, pp1q psq “ a, pp1q ptq “ b . As a function on Tp1q ˆ Tp1q , D1˝ needs not satisfy the triangle inequality. This motivates one to define, for a, b P Tp1q , ˜ (1.1.1)

D1˚ pa, bq “ inf

k´1 ÿ i“1

¸ D1˝ pai , ai`1 q : k ě 1, a “ a1 , a2 , . . . , ak´1 , ak “ b

.

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

It follows that the function D1˚ is a pseudo-metric on Tp1q ˆ Tp1q ; so we may have D1˚ pa, bq “ 0 for a ‰ b. See [26, Exercise 3.1.13] or [48, Section 1]. Now write M “ Tp1q {tD1˚ “ 0u , and let ρ1 be the equivalence class in M of pp1q p0q. Write π1 for the canonical projection from Tp1q to M, sending a P Tp1q to its equivalence class in M. We continue to use D1˚ to denote the push-forward of D1˚ by pπ1 , π1 q to M ˆ M. Finally, let LebR be the Lebesgue measure on R, and let µ1 “ pπ1 ˝ pp1q q˚ LebR . Then µ1 is a uniform probability measure on M. We define m8 “ pM, D1˚ , ρ1 , µ1 q to be the pointed measured Brownian map. This definition may be confusing to a reader encountering it for the first time. It is a continuum analogue of a family of bijections from labelled trees to maps. We describe one such bijection, the BDG bijection, in Section 2.2.3; it may be useful to revisit the current section after reading the description of the BDG bijection. 1.1.4. The Brownian Map as the Unique Limit. In this subsection, we briefly overview the results of (1) convergence of 2p-angulations to the Brownian map; (2) uniqueness of the Brownian map; and (3) generalization to convergence of triangulations. (1) Convergence of 2p-angulations to the Brownian map. Similar to the methodologies for deducing radius and profiles of large random maps, Le Gall’s proof for Theorem 1.1.1 also makes use of bijections between labelled trees and maps. In particular, the paper [48] uses the BDG bijection by Bouttier, Di Francesco & Guitter [25] to uniquely encode a 2p-angulation with a labelled planted

10

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

plane tree. See Section 2.2 for a detailed description of labelled planted plane trees, contour exploration, contour process, label process, the BDG bijection, et cetera. Under such a bijection, Le Gall is able to control the distances from the distinguished vertex rather than from the root. He accomplished this by adapting the proof of [46, Theorem 3.4] from rooted quadrangulations to pointed and rooted quadrangulations, leading to the joint convergence of [48, Theorem 2.3] for continuous processes along a subsequence. We need to introduce a bit more notation before presenting the theorem. A reference to Section 2.2.3 might be helpful if the reader encounter the BDG bijection for the first time. Fix p P N with p ě 2. For each n P N, let Mn be a uniform pointed and rooted 2p-angulation with n faces, pointed at the vertex Bn ; Bn corresponds to the extra vertex added in the closure operation of the BDG bijection. Let Tn be the labelled planted plane tree corresponding to Mn under the BDG bijection. Let Cn “ pCn ptq : 0 ď t ď 1q be the contour process of Tn and let Zn “ pZn ptq : 0 ď t ď 1q be the label process of Tn . Write pv0 , . . . , v2pn q for the sequence of vertices visited by Cn , with v0 “ v2pn ; note that there are pn vertices in Tn according to the BDG bijection. Next, for i, j P t0, . . . , 2pnu, let dn pi, jq “ dMn pvi , vj q, and use linear interpolation to extend the domain of dn to r0, 2pns2 . Furthermore, Le Gall & Weill [51, Proposition 2.5] showed that there is an almost surely unique instant s˚ P p0, 1q such that Zs˚ “ min Zs , 0ďsď1

recalling the definition of Z from Section 1.1.3.

11

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

Theorem 1.1.3 ([48, Theorem 2.3]) Write 1 λp “ 2

c

p , κp “ p´1

ˆ

9 4ppp ´ 1q

˙1{4 .

For every sequence of integers converging to 8, we can extract a subsequence pnk qkPN along which the following convergence in distribution of continuous processes holds: (1.1.2) ˆ ˙ λp κp κp d ? Cn ptq, 1{4 Zn ptq, 1{4 dn p2pns, 2pntq Ñ pet , Zt , Dps, tqq0ďs,tď1 n n n 0ďs,tď1 as n Ñ 8, where pDps, tqq0ďs,tď1 is a continuous random process such that the function ps, tq Ñ Dps, tq defines a pseudo-metric on r0, 1s2 , and the following properties hold: Dps, s˚ q “ Zs ´ Zs˚ “ D1˝ ppp1q psq, pp1q ps˚ qq for every s P r0, 1s ; Dps, tq ď D1˚ ppp1q psq, pp1q ptqq ď D1˝ ppp1q psq, pp1q ptqq for every s, t P r0, 1s . For every s, t P r0, 1s, we put s « t if Dps, tq “ 0. Then, almost surely for every s, t P r0, 1s, the property s « t holds if and only if D1˚ ppp1q psq, pp1q ptqq “ 0, or equivalently D1˝ ppp1q psq, pp1q ptqq “ 0.

Finally, set M “ r0, 1s{ « and equip M with the distance induced by D, which is still denoted by D. Then, along the same sequence where the convergence (1.1.2) holds, `

˘ d vpMn q, κp n´1{4 ¨ dMn Ñ pM, Dq

as n Ñ 8 in distribution in the sense of the Gromov-Hausdorff convergence. (2) Uniqueness of the Brownian map. In Theorem 1.1.3, the only thing that remains to show for the uniqueness of the Brownian map is that D does not not depend on the subsequence. The uniqueness of the Brownian map relies on the property that there is an almost surely unique

12

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

geodesic path between any two points in the Brownian map, shown by Le Gall [47]. This leads to the next theorem, which characterizes the metric space pM, D1˚ q where D1˚ is given in (1.1.1), and we continue to use D1˚ to denote the push-forward onto M ˆ M. Theorem 1.1.4 ([48, Theorem 7.2]) We have Dpy, y 1 q “ D1˚ py, y 1 q for every y, y 1 P M, almost surely.

Theorem 1.1.4 tells us that the limiting space pM, Dq in Theorem 1.1.3 coincides with pM, D1˚ q, almost surely. Then the convergence of Theorem 1.1.3 does not depend on the choice of the subsequence pnk qkPN , so this completes the proof of uniqueness. The proof for Theorem 1.1.4 constitutes the majority of the paper [48], and we refer the reader to that work for a complete exposition. (3) Generalization to convergence of triangulations. Le Gall showed that, once convergence for a single map model is known, this may be used to “bootstrap” convergence proofs for other models. More specifically, he used the following distribution identities in order to describe a “re-rooting invariance trick”. Theorem 1.1.5 ([48, Corollary 7.3]) Let U and V be two independent random variables uniformly distributed over r0, 1s, and such that the pair pU, V q is independent of pe, Zq, where the pair pe, Zq is defined in Section 1.1.3. Then we have d

d

D1˚ ppp1q pU q, pp1q pV qq “ D1˚ ppp1q ps˚ q, pp1q pU qq “ ZU ´ min Zs “ ´ min Zs . 0ďsď1

0ďsď1

Using this theorem, Le Gall shows that convergence in the case q “ 3 is a consequence of convergence for q “ 4, together with the following facts. For v P vpMn q, write tv P r0, 1s for the last time v is visited by the contour exploration on Tn , recalling that Tn is the labelled tree corresponding to Mn under the BDG bijection. The remaining main ingredients to prove the convergence for q “ 3 are given below.

13

1.1. THE BROWNIAN MAP AS A SCALING LIMIT OF RANDOM MAPS

(1) Distances to the minimal point: For v P vpMn q, dMn pv, Bn q “ Zn ptv q ´ minpZn ptq : 0 ď t ď 1q ` 1 ; see [48, (3)]. (2) Bounding the distance between any two vertices: For any v, u P vpMn q with tv ď tu , ˆ dMn pv, uq ď Zn ptv q ` Zn ptu q ´ 2 max

min Zn ptq,

tPrtv ,tu s

min tPr0,tv sYrtu ,1s

˙ Zn ptq ` 2 ;

see [48, (4)]. A systematic presentation of how to use the re-rooting invariance trick to prove convergence is described by Addario-Berry & Albenque [4, Theorem 4.1].

1.1.5. Universality of the Brownian Map. In this subsection, we provide a quick account of recent progress on showing the universality of the Brownian map. Bettinelli showed in [21] that large bipartite quadrangulations of positive genus g rescale, regardless of the choice of the subsequence, to a limiting space which is homeomorphic to the genus g-torus, and in [22] that the Brownian map is again the scaling limit of large quadrangulations with a boundary whose length is of order the square root of the number of faces. Bettinelli, Jacob & Miermont [23] proved that uniform rooted plane maps with a fixed number of edges also rescale to the Brownian map; their proof relies on the Ambj∅rn-Budd bijection [11]. Beltran & Le Gall [17] proved that uniform quadrangulations without pendant vertices rescale to the Brownian map as well. Addario-Berry & Albenque [4] showed that the scaling limit of simple triangulations and quadrangulations is again the Brownian map. Janson & Stef´ansson [42] showed that bipartite maps with a unique large face under Boltzmann distribution also rescale to the Brownian map; their proof relies on the BDG bijection and recent results on simply generated trees where a unique vertex

14

1.2. THE BROWNIAN PLANE

of a high degree appears when the trees are large. The latter result is also related to Chapter 4 of this thesis.

1.2. The Brownian Plane as a Non-Compact Limit of Random Maps We have seen that after properly rescaling the graph distance, random maps converge in the Gromov-Hausdorff sense to the Brownian map, which is a random compact metric space. Note that rescaling is necessary for obtaining a compact limit. On the other hand, we may study convergence of random maps without rescaling the distance. In this case we have to consider other topology rather than the Gromov-Hausdorff topology. Several authors consider the local limits for graphs in the following sense. Given a graph G and v P vpGq, for k P N, write Bk pv; Gq for the subgraph of G induced by the vertex set tu P vpGq : dG pv, uq ď ku. A pointed graph is a pair pG, ρq where G is a graph and ρ P vpGq. A sequence pGn , ρn q of pointed graphs is said to converge locally to a pointed graph pG, ρq if for every k P N, for every large enough n, Bk pρn ; Gn q “ Bk pρ; Gq, where the equality is considered up to isomorphism between finite pointed graphs. Angel & Schramm [14] proved the existence of the local limit of uniform finite rooted triangulations, and called the limit object the uniform infinite planar triangulation (UIPT). Subsequently, Angel [13] gave a construction as a growth process for sampling the UIPT. Later, Krikun [43] established local limit of uniform finite rooted quadrangulations, called the uniform infinite planar quadrangulation (UIPQ). Chassaing & Durhuus [27] studied the limit of uniform probability measure over a family of finite labelled trees, then used the Schaeffer’s bijection [62] to relate that limit object to uniform measure over the set of infinite planar quadrangulations. M´enard [55] proved that the limiting uniform measures studied in [43] and in [27] are the same. Gurel-Gurevich & Nachmias [39] showed that simple random walk on the UIPT or the UIPQ is recurrent. Benjamini & Curien [19] proved that simple random walk on the UIPQ is subdiffusive.

15

1.2. THE BROWNIAN PLANE

16

Recently, Curien & Le Gall [30] made a connection between the scaling towards the Brownian map and the local convergence towards the UIPQ. They showed that upon multiplying the graph distance on the UIPQ by a factor tending to 0, the resulting metric spaces converge to the Brownian plane. We give a definition of the Brownian plane in Section 1.2.1. Subsequently, Curien & Le Gall [31] introduced a hull process of the Brownian plane and deduced more properties of it. The same authors [32] also studied the scaling limits of the volumes and perimeters of the discovered regions in Markovian explorations (also known as peeling processes) of UIPQ and UIPT. We review the main results of [30] in Section 1.2.2, referring the interested reader to [31, 32] for subsequent works.

1.2.1. The Brownian Plane. We quickly go over the definition of the Brownian plane from [30], referring the reader to that work for a full exposition. Like with the construction of the Brownian map, the following construction may be better understood after reading the description of the BDG bijection in Section 2.2.3. Let R “ pRt qtě0 and R1 “ pRt1 qtě0 be two independent three-dimensional Bessel processes started from 0. We define R “ pRt qtPR by setting $ ’ ’ &Rt if t ě 0 Rt “ ; ’ ’ 1 %R´t if t ă 0 the construction is analogous to the Brownian map but we use R rather than Brownian excursion e. Then for any s, t P R, let $ ’ ’ &rs ^ t, s _ ts st “ ’ ’ %p´8, s ^ ts Y rs _ t, 8q

if st ě 0 , if st ă 0

1.2. THE BROWNIAN PLANE

17

and define a process Z “ pZt qtPR such that, conditional on R, Z is the centred Gaussian process with covariance E rZs Zt | Rs “ inf Rr rPst

for any s, t P R. Next, define a random pseudo-metric dR on R2 by setting, for any s, t P R, dR ps, tq “ Rs ` Rt ´ 2 inf Rr . rPst

Write s „R t if dR ps, tq “ 0, and let T8 “ R{ „R . We call T8 the infinite Continuum Random Tree. Write p8 : R Ñ T8 for the canonical projection sending x P R to its equivalence class in T8 . Furthermore, for any s, t P R, let (1.2.1)

˝ D8 ps, tq “ Zs ` Zt ´ 2

inf rPrs^t,s_ts

Zr .

˝ Then we extend the domain of definition for D8 to T8 ˆ T8 by setting, for any

a, b P T8 , ˝ ˝ D8 pa, bq “ min tD8 ps, tq : s, t P R, p8 psq “ a, p8 ptq “ bu .

Now, let (1.2.2)

D8 pa, bq “

inf

a0 “a,a1 ,...,ak “b

k ÿ

˝ D8 pai´1 , ai q

i“1

with the infimum taken over all choices of k P N and of the finite sequence a0 “ a, a1 , . . . , ak “ b P T8 . It follows that D8 is a pseudo-metric on T8 ; see [30, Section 1]. Write P “ T8 {tD8 “ 0u

1.2. THE BROWNIAN PLANE

and let ρ8 be the equivalence class of p8 p0q in P. We continue to use D8 to denote the push-forward of D8 to P ˆ P. Finally, write π8 : T8 Ñ P for the canonical projection, sending a P T8 to its equivalence class in P. Let µ8 “ pπ8 ˝ p8 q˚ LebR . Then write P “ pP, D8 , ρ8 , µ8 q for the pointed measured Brownian plane.

1.2.2. Properties of the Brownian Plane. The pointed Brownian plane pP, D8 , ρ8 q is a non-compact random pointed metric space, so the Gromov-Hausdroff (GH) topology is too strong for convergence. We consider a slight generalization of the GH topology, called the local GH topology, explained as follows. A pointed metric space is a triple pV, d, oq where pV, dq is a metric space and o P V . Given a pointed metric space V “ pV, d, oq, for any r ě 0, let Br pVq “ tv P V : dpv, oq ď ru, and write Br pVq “ pBr pVq, d, oq. Informally, a sequence of pointed metric spaces pVn : n P Nq converges to V in the local GH topology if for any r ě 0, Br pVn q converges to Br pVq in the pointed GH topology. See Section 4.2 for more detailed definitions of the pointed and the local GH topologies. Curien & Le Gall [30] showed that the Brownian plane can be viewed as the Gromov-Hausdorff tangent cone in distribution to the Brownian map at its uniform point.

18

1.2. THE BROWNIAN PLANE

Theorem 1.2.1 ([30, Theorem 1]) For every δ ą 0, we can find ε ą 0 and construct on the same probability space copies of the Brownian plane P and of the Brownian map m8 , in such a way that the balls Bε pPq and Bε pm8 q are isometric with probability at least 1 ´ δ. Furthermore, we have pM, λ ¨ D1˚ , ρ1 q Ñ pP, D8 , ρ8 q in distribution for the local Gromov-Hausdorff topology as λ Ñ 8.

This theorem can be easily generalized to the convergence in the local GromovHausdorff-Prokhorov topology, which is what this thesis indeed uses; see Section 4.11. It follows from Theorem 1.2.1 that the Brownian plane has the scale invariance property, that is, for every λ ą 0, pP, λ ¨ D8 , ρ8 q has the same distribution as pP, D8 , ρ8 q. This allows more tractable calculations of some explicit distributional properties of the Brownian plane, the analogues of which are more difficult to obtain for the Brownian map. Next, we present another theorem of [30] which links the UIPQ to the Brownian plane. Let Q8 be the UIPQ, and denote by ρp8q the distinguished point (also called the root) of Q8 .

19

1.3. SCALING LIMITS OF MAPS WITH HIGHER CONNECTIVITY

Theorem 1.2.2 ([30, Theorem 2]) We have pvpQ8 q, λ ¨ dQ8 , ρp8q q Ñ pP, D8 , ρ8 q as λ Ñ 0 in distribution for the local Gromov-Hausdorff topology. Furthermore, let pkn qnPN be a sequence of non-negative real numbers such that kn Ñ 8 and kn “ opn1{4 q as n tends to infinity. Let Qn be a uniform pointed and rooted quadrangulation with n faces and with the distinguished point ρpnq . Then, pvpQn q, kn´1 ¨ dQn , ρpnq q Ñ pP, D8 , ρ8 q as n Ñ 8 in distribution for the local Gromov-Hausdorff topology.

1.3. Scaling Limits of Maps with Higher Connectivity Connectivity of planar graphs is a research topic with a long history and it has ramifications in many other branches of science. In the groundbreaking work by Tutte [65], enumeration of several families of maps were made possible by recursively decomposing maps into components of higher connectivity. Extending these results, Gao & Wormald [37] derived sizes of the largest components in random maps, and Banderier, Flajolet, Schaeffer & Soria [16] gave detailed distributional information on the sizes, using singularity analysis and saddle point method. This line of research on map enumeration relies heavily on expressing decomposition of maps with generating functions, and counting the number of maps in a given family by extracting coefficient of the corresponding generating function. Building principally on the works of Le Gall [48], Miermont [59], Addario-Berry & Albenque [4], Curien & Le Gall [30], Banderier, Flajolet, Schaeffer & Soria [16], Gao & Wormald [37], and Tutte [65], this thesis further investigates the critical and subcritical scaling limits of random quadrangulations with connectivity constraints. We present a quick overview of these topics in the next two subsections, deferring the details to Chapters 3 and 4, respectively.

20

1.3. SCALING LIMITS OF MAPS WITH HIGHER CONNECTIVITY

1.3.1. The Scaling Limit of 2-Connected Quadrangulations. A graph is 2-connected if the removal of any vertex does not disconnect the graph. A 2connected block in a graph is a maximal 2-connected induced subgraph. A graph is simple if there are no multiple edges or loops. A simple block in a graph is a maximal simple induced subgraph. We elaborate on this definition, in the setting of maps, in Section 3.1. In Chapter 3, which is based on the manuscript [5], we show that a uniform quadrangulation, its largest 2-connected block, and its largest simple block jointly rescale to the same Brownian map in distribution for the Gromov-Hausdorff-Prokhorov topology. As a start, a local limit theorem is derived for the asymptotics of maximal block sizes, extending the result by Banderier, Flajolet, Schaeffer & Soria [16]. More precisely, we show that with high probability, the largest 2-connected block in a uniform rooted quadrangulation of size n has

7 n ` Opn2{3 q 15

vertices, and the largest

simple block in a uniform rooted 2-connected quadrangulation of size n has 57 n ` Opn2{3 q vertices. Those results are obtained by applying Tutte’s quadratic method (Section 3.9) and the mechanism of singularity analysis developed for maps with higher connectivity constraints (Section 3.3). We thereby obtain size and diameter bounds for submaps pendant to random quadrangulations. To deduce the scaling limit of 2-connected quadrangulations, we apply Theorem 1.1.1 together with a theorem by Addario-Berry & Albenque [4], stated below; we only use their result for simple quadrangulations, but we also state the triangulation case for completeness. Given a finite graph G, write µG for the uniform probability measure over vpGq. A measured metric space is a triple pV, d, µq where pV, dq is a metric space and µ is a finite non-negative Borel measure on V . Recall that pM, D1˚ , µ1 q denotes the measured Brownian map. We assume familiarity with Gromov-Hausdorff-Prokhorov topology, and refer the reader to Section 3.2.2 for a quick review.

21

1.3. SCALING LIMITS OF MAPS WITH HIGHER CONNECTIVITY

Theorem 1.3.1 ([4, Theorem 1.1]) For n ě 4, let Qn be a uniformly random element of the set of rooted simple quadrangulations with n vertices, and let Tn be a uniformly random element of the set of rooted simple triangulations with n vertices. Then ˜ ¸ ˆ ˙1{4 8 vpQn q, ¨ dQn , µQn Ñ pM, D1˚ , µ1 q 3n and

˜ vpTn q,

ˆ

4 3n

¸

˙1{4 ¨ dTn , µTn

Ñ pM, D1˚ , µ1 q

as n Ñ 8 in distribution for the Gromov-Hausdorff-Prokhorov topology.

We show that the joint convergence of random quadrangulations and their cores follows from Theorems 1.1.1 and 1.3.1. The basic idea of the proof is that, the largest 2-connected block is a “sandwich” between two metric spaces which jointly converge to the same limit object after rescaling the graph distances.

1.3.2. Subcritical Scaling Limit of Quadrangulations Conditional on Having Large Root Block. The 2-connected root block of a rooted quadrangulation is the 2-connected block that contains the root edge. Let r : N Ñ N be such that rpnq ą pln nq25 for all n and rpnq “ opnq as n Ñ 8. For n P N, let Qn be a uniform rooted measured quadrangulation of size n with rpnq vertices in the 2-connected root block. Chapter 4, which is based on the ´ ¯1{4 21 , with manuscript [68], shows that, upon rescaling the graph distance by 40rpnq an appropriate rescaling of measure, Qn converges to a limiting space obtained by identifying a uniform point of the Brownian map with the distinguished point of the Brownian plane. We briefly describe the proof idea. The convergence result of Chapter 3 entails that the root block of Qn rescales to the Brownian map. Furthermore, for each n P N, write Λn for the size of the largest component ΛpQn q in the complement of the root block. We show that lim inf nÑ8

Λn rpnq

“ 8 with high probability. Furthermore, since

22

1.4. NOTATION

23

rpnq “ opnq and rpnq Ñ 8 as n Ñ 8, it follows from Theorem 1.2.2 that ΛpQn q converges, upon rescaling the graph distance by c ¨ rpnq´1{4 for some c ą 0, to the Brownian plane. Finally, we show that the rest of the components are uniformly asymptotically negligible, which vanishes in the scaling limit. Thus, the limiting object may be viewed as the Brownian map attached to the Brownian plane. To deduce the asymptotic sizes of ΛpQn q and other components, we use a random allocation model, and generalize existing results on “condensation” in such a framework. 1.4. Notation Throughout the thesis, we use the following notation. Given a set of graphs G, for n P N, let Gn “ tG P G : |vpGq| “ nu. Given a finite set G, the notation d

G Pu G means that G is chosen uniformly at random from G. We denote by Ñ p

and Ñ convergence in distribution and in probability, respectively. When we say that a sequence pEn : n P Nq of events occurs with high probability, we mean that P pEn q Ñ 1 as n Ñ 8. Finally, we write N “ t1, 2, . . .u and Ně0 “ t0, 1, . . .u.

Several Concepts about Maps

25

CHAPTER 2

Several Concepts about Maps 2.1. Graphs and Maps In this section, we review relevant definitions of graphs, planar graphs, and maps, paving the way for the study to come.

2.1.1. Graphs. We provide a brief overview of graphs, referring the reader to the exposition [24] by Bollob´as for greater details. A graph G is an ordered pair of disjoint sets pvpGq, epGqq together with a function fG : epGq Ñ vpGq Y vpGq2 , where vpGq2 denotes the set of unordered pairs of vpGq. We call vpGq the set of vertices of G and epGq the set of edges of G. Unless stated otherwise, we assume that vpGq is finite. For ease of presentation, we often omit fG from notation and represent epGq as a subset of vpGq2 . Though writing an edge e “ tx, yu P vpGq2 does not determine e uniquely, it rarely causes confusion in the context of this thesis. Given a graph G, if tu, vu P epGq, we say that tu, vu joins the vertices u and v. We call u and v the extremities, or endpoints, of tu, vu, and we say that tu, vu is incident to u and v. Two vertices are adjacent if they are joined by an edge. Two edges are adjacent if they have exactly one common endpoint. The degree of a vertex v in a graph G is the number of edges incident to v in G, denoted by degG pvq; a loop is counted as 1 degree. Multiple edges in a graph are distinct edges with the same pair of distinct extremities. Loops in a graph are edges with only one end. Simple graphs are graphs without multiple edges and loops. Some authors use the term multigraphs for graphs with multiple edges or loops, and reserve the term graphs for simple graphs; we do not use this convention here.

2.1. GRAPHS AND MAPS

26

A graph is 2-connected if the removal of any vertex does not disconnect the graph.

We say that G1 “ pV, Eq is a subgraph of G if V Ă vpGq and E Ă epGq. Furthermore, if G1 contains all edges of G that join two vertices in V , then G1 is said to be the subgraph induced by V and is denoted by GrV s. We write G ´ V “ GrvpGqzV s.

An oriented edge of G is an ordered pair pu, vq, denoted by uv, where tu, vu P epGq. A rooted graph is a pair pG, uvq where G is a graph and uv is an oriented edge of G. If G has a single vertex, say v, then we abuse notation and say that pG, vq is the corresponding rooted graph.

2.1.2. Planar Graphs. Basic concepts of planar graphs are given here. For a complete treatment, please see Mohar & Thomassen [61]. Let X be a topological space. A curve, or edge, in X is the image of a continuous function f : r0, 1s Ñ X, and we say that the curve is simple if either f is one-to-one, or the restriction of f to r0, 1q is one-to-one and f p0q “ f p1q (a loop). We call f p0q and f p1q the endpoints of the curve f pr0, 1sq, and f pr0, 1sq is said to join the endpoints. The interior of a curve e “ f pr0, 1sq is the set f pp0, 1qq, denoted by intpeq.

An embedded graph in X is a graph G such that (1) vpGq are finitely many distinct elements of X; (2) for all e P epGq, e is a simple curve joining the endpoints in X which are extremities of e in G; and (3) for all e P epGq, intpeq does not intersect other edges and vertices. The support of an embedded graph G is the set supppGq “ vpGq Y

ď

intpeq ,

ePepGq

and a face of G is a connected component of XzsupppGq.

2.1. GRAPHS AND MAPS

An embedding of a graph G in X is an embedded graph G1 in X such that G and G1 are isomorphic, and G1 is said to be a representation of G in X. We say that a graph can be embedded into X if there is a representation in X. A graph is planar if it can be embedded in the Euclidean plane R2 . All graphs considered in the thesis are planar.

2.1.3. Maps. Some basic concepts related to planar maps were described at the start of Chapter 1. We give a few additional definitions here. Let X be a topological space. A map on X is an embedded graph in X whose faces are all homeomorphic to the unit disk of R2 . Note that for an embedded graph in a surface of higher genus, faces are not necessarily homeomorphic to the unit disk of R2 . A plane graph is an embedded graph in the plane R2 . A plane map or planar map is an embedded graph in the 2-sphere S2 . The point set of a plane graph G is compact, so exactly one face of G (i.e. R2 zG) is unbounded, and we call it the unbounded face of G. For the duration of this thesis, all maps are embedded in either R2 or S2 and they are considered up to orientation-preserving homeomorphism. In the case that only the metric structure of a map is relevant, we may call it a graph. From now on, we only consider graphs and maps without loops. A corner of a map M is a pair pe, e1 q where e and e1 are adjacent edges of M such that e1 follows e in the clockwise order around the shared vertex. We recall from Chapter 1 that the degree of a face is the number of edges incident to the face, and that for integer q ě 3, a q-angulation is a map in which every face has degree q. In particular, for q “ 3 and q “ 4, respectively, we call such maps triangulations and quadrangulations.

2.1.4. Total Orders on Maps. For this subsection, a reference to Figure 2.1 may be helpful. A rooted map is a pair pM, uvq where M is a map and uv is an oriented edge in M . Given a rooted map M “ pM, uvq, we may define a canonical total order ăM

27

2.2. LABELLED TREES AND THE BDG BIJECTION

11 112 H

1 1111

2

12

111

Figure 2.1. pH, 1q is the root edge of M. For the total order ă“ăM we have, e.g., pH, 1q ă pH, 2q, p2, 12q ă p12, 2q ă p12, 111q ă p111, 12q. Also, of the two copies of edge p11, 112q, the one succeeding p11, 2q in the clockwise order is smaller for ă.

on vpM q as follows. List the vertices of M as u1 “ u, u2 “ v, . . . , u|vpM q| according to their order of exploration by a breadth-first search (see [34] for the definition of breadth-first search) which starts from the root edge uv and uses the clockwise order of edges around each vertex starting from the explored edge to determine exploration priority. We also define a total order ăM on the set of oriented edges of M as follows. Let ui uj ăM ui1 uj 1 precisely if either (a) ui was explored before ui1 or (b) i “ i1 and ui uj has higher exploration priority than ui uj 1 .

2.2. Labelled Trees and the BDG Bijection 2.2.1. Trees. A tree is a connected acyclic graph. A plane tree is a tree embedded in the plane. A planted plane tree is a rooted map pT, uvq where T is a tree. If a planted plane tree pT, uvq has only one vertex u, uv denotes the vertex u. Fix a planted plane tree T “ pT, uvq. We view a planted plane tree T as (vertex) rooted at u, which allows us to speak of children, parents, ancestors, et cetera. Given w P vpT qztuu and p P vpT q such that dT pw, pq “ 1 and dT pu, wq ´ dT pu, pq “ 1, then we say p is the parent of w, and w is a child of p. The contour exploration of T is a function r “ rT : t0, . . . , 2|epT q|u Ñ vpT q defined as follows. Let rp0q “ u; then for each i “ 1, . . . , |epT q|, let rpiq be the lexicographically (i.e. ăT -order) first child of rpi ´ 1q that is not an element of

28

2.2. LABELLED TREES AND THE BDG BIJECTION

trp0q, . . . , rpi ´ 1qu, or let rpiq be the parent of rpi ´ 1q if no such vertex exists. We sometimes refer rpiq as a corner. Note that there may be several corners corresponding to a vertex. See Figure 2.2.2 (a) for an example.

2.2.2. Contours and Labels for p-Trees. Bijections between labelled trees and maps allow us to enumerate many different families of maps. Highlights in the bijective theory of maps enumeration include the CVS bijection by Cori & Vauquelin [29] and Schaeffer [62], the BDG bijection by Bouttier, Di Francesco & Guitter [25], the unified bijection by Bernardi & Fusy [20]. The common thread is that all these works associate families of maps and families of labelled trees via some sort of folding/unfolding operation. We describe in the next subsection a version of the BDG bijection which is used to prove the convergence to and the uniqueness of the Brownian map by Le Gall [48] in the case of 2q-angulations for q ě 2. In order to do so, we first define p-trees, and their contour and label processes. For other description of these objects, we refer the reader to [48, Section 2.1]. Given a planted plane tree T “ pT, uvq and a vertex w P vpT q, write kT pwq for the number of children of w in T. If kT pwq ‰ 0 then list the children of w in lexicographic order (i.e. ăT -order) as kT pw, 1q, . . . , kT pw, kT pwqq. If w ‰ u, write kT pw, 0q for the parent of w. For the remaining subsection, a reference to Figure 2.2.2 may be helpful. Fix p P N with p ě 2. A p-tree is a plane tree T “ pT, uvq such that for every w P vpGq with dT pw, uq being odd, we have kT pwq “ p ´ 1. Fix a p-tree T “ pT, uvq. Let WT be the set of vertices w P vpT q such that dT pw, uq is even, and call the elements of WT white vertices. Let BT “ vpT qzWT , and call the elements of BT black vertices. In the BDG bijection, black tree vertices correspond to map faces, and white tree vertices correspond to map vertices. A function X “ XT : WT Ñ Z is a valid labelling of T if the following properties hold: (1) Xpuq “ 0;

29

2.2. LABELLED TREES AND THE BDG BIJECTION

30

(2) given w P BT , for each i “ 0, . . . , p ´ 1, we have XpkT pw, i ` 1qq ě XpkT pw, iqq ´ 1 , where kT pw, pq – kT pw, 0q is the parent of w. We call XT pvq the label of v for v P vpT q. A labelled p-tree is a triple pT, uv, Xq, where pT, uvq is a p-tree and X is a valid labelling of pT, uvq. We write W p for the set of labelled p-trees. For any n P N, let Wnp “ tpT, uv, Xq P W p : |BpT,uvq | “ nu. Fix a labelled p-tree θ “ pT, uv, Xq P Wnp , then |epT q| “ pn. Let r “ rpT,uvq be the contour exploration of pT, uvq. By parity, for all i “ 0, . . . , pn, rp2iq is a white vertex. The contour process C “ Cθ : r0, 1s Ñ R and the label process Z “ Zθ : r0, 1s Ñ R of θ are defined as follows. First, for each i “ 0, . . . , pn, let ˆ C

i pn

˙

1 “ dT pu, rp2iqq 2

and ˆ Z

i pn

˙ “ Xprp2iqq .

Then extend the domain of each function to r0, 1s by linear interpolation. 2k

1k

@ @w @

0k

@ @w @k 3 @ @w

rp2q 0k

1k

2k

rp4q 1k 2k @ rp3q @ @w @w rp1q Zrp5q  Z  Z rp6q  Z  rp0q 0k

1k

Figure 2.2.2 (a). A labelled 3-tree θ with 5 black vertices. r is its corresponding contour exploration. For cleanness, only rp0q, . . . , rp6q are shown here.

2.2. LABELLED TREES AND THE BDG BIJECTION

Cθ 6

r C C C

r      r

r C C

r    

C C Cr

31

C C C Cr C C

 r

   r C C

r

1   

C C

  r

r C C C

C Cr

  C

  r

 C  C  Cr

C C Cr-

1 t

1 15

Figure 2.2.2 (b). The contour process Cθ of the labelled 3-tree θ. Zθ 6 r B  B r   r B  B

1 r

r



    B  B r

1 15



B

r B  B  B Br  B  B 

Br B B B  B r

r  B

 

Br B

  

B B  B r

r B B B

Br B B B Br

1 t

Figure 2.2.2 (c). The label process Zθ of the labelled 3-tree θ.

2.2.3. The BDG Bijection.

2.2. LABELLED TREES AND THE BDG BIJECTION

32

Bk

2k

1k

1k

@ @w @

@ @w @k 3 @ @w

0k

0k

1k

@ @w Z Z Z

2k

2k

Z k  0

1k

@ @w   

Figure 2.2.3. A labelled 3-tree and its associated 6-angulation, recalling that if both sides of an edge are incident to the same face then this edge is counted twice.

Fix p P N with p ě 2. For n P N write Mpn for the set of rooted and pointed 2p-angulations with n faces. We give a brief introduction to the BDG bijection from Wnp ˆ t0, 1u to Mpn , referring the reader to [48] for more details and a description of the inverse bijection. For the remainder of this subsection, a reference to Figure 2.2.3 may be useful. Fix θ “ pT, uv, Xq P Wnp and ε P t0, 1u. Recall from Section 2.2.1 that r “ rpT,uvq : t0, . . . , 2pnu Ñ vpT q is the contour exploration of pT, uvq. We extend the domain of r to t0, . . . , 4pnu by letting rp2pn ` iq “ rpiq for i “ 0, . . . , 2pn. Since T is a plane tree, we may view it as embedded on the 2-sphere S2 . Add an extra vertex B on the sphere where T is embedded. Recall that WpT,uvq is the set of white vertices of pT, uvq. We then associate with pθ, εq a 2p-angulation M of S2

2.2. LABELLED TREES AND THE BDG BIJECTION

33

with n faces, whose vertex set is WpT,uvq Y tBu, and whose edge set is obtained as follows: for each i “ 0, . . . , pn ´ 1,

(i) if Xprp2iqq “ minpXpzq : z P WpT,uvq q, draw an edge between the corner rp2iq and B; (ii) otherwise, draw an edge between the corner rp2iq and rp2jq, where j is the first index in the sequence i ` 1, . . . , i ` pn ´ 1 such that Xprp2jqq “ Xprp2iqq ´ 1 . Note that condition (2) in the definition of valid labellings implies that for i P t1, . . . , pnu, Xprp2iqq ě Xprp2i ´ 2qq ´ 1, so such an index j in (ii) must exist; see Figure 2.2.2 (c) for an example. The resulting map M is a 2p-angulation, and each face of M contains exactly one black vertex of T ; see [25]. If minpXpzq : z P WpT,uvq q “ 0 then root M at the edge between B and the corner rp0q, oriented from B to rp0q if ε “ 1 or the other way around if ε “ 0; otherwise, root M at the edge between the corners rp0q and rp2jq where j is the first index in the sequence 1, . . . , pn such that Xprp2jqq “ ´1, oriented from rp2jq to rp0q if ε “ 1 or the other direction if ε “ 0. Let M be pointed at the vertex B. An important property of BDG bijection is that the labels on WpT,uvq give information about distances to B in M . More precisely, for all edges tv, wu of M , |Xpvq ´ Xpwq| “ 1, and for all v ‰ B there is a neighbour w of v such that Xpwq “ Xpvq ´ 1. It follows that for w P WpT,uvq , (2.2.1)

dM pB, wq “ Xpwq ´ minpXpzq : z P WpT,uvq q ` 1 .

For i, j P t1, . . . , pn ´ 1u with i ă j, there is in general no simple expression for the exact value of dM prp2iq, rp2jqq in terms of the labels, but we do have the bound (2.2.2) ˆ dM prp2iq, rp2jqq ď Xprp2iqq`Xprp2jqq´2 max

˙ min Xprp2kqq,

iďkďj

min

jďkďi`pn

Xprp2kqq `2 .

2.3. MAP ENUMERATION AND SINGULARITY ANALYSIS

We briefly explain why this holds; see also [48, Section 2.2]. Fix isint1, . . . , pn ´ 1u. For every integer k P Ně0 with k ď Xprp2iqq ´ minpXpzq : z P WpT,uvq q, let φpiq pkq “ min pj P ti, . . . , i ` pn ´ 1u : Xprp2jqq “ Xprp2iqq ´ kq , and ωpiq pkq “ rp2φpiq pkqq. Informally, ωpiq pkq is the first vertex after rp2iq in the clockwise order of the contour exploration such that the label of ωpiq pkq is k smaller than rp2iq. Again, condition (2) in the definition of valid labelling implies that such vertex must exist. Next, we let ωpiq pdM prp2iq, Bqq “ B. It then follows from (2.2.1) that `

˘ ωpiq pkq : 0 ď k ď dM prp2iq, Bq

is a discrete geodesic from rp2iq to B in M . Such a geodesic is called a (discrete) simple geodesic. Finally, note that dM prp2iq, rp2jqq is at most the length of the path obtained by concatenating the simple geodesics φpiq and φpjq up to their coalescence time. Such kind of bound on the distance between any two vertices is essential in deducing the convergence to the Brownian map, as discussed in Section 1.1.4.

2.3. Map Enumeration and Singularity Analysis An important facet of the study of maps is map enumeration, which often plays a primary role in solving random map problems. In Section 2.2, we have seen a bijection between labelled trees and maps, which entails us to count maps by counting labelled trees. Another similar technique is to establish bijections between blossoming trees and maps, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable reconstruction of its faces. A key example in the latter category is the work of Albenque & Poulalhon [6], who defined a unified bijective scheme between planar maps and blossoming trees. The map enumeration techniques that we use in the upcoming chapters are the quadratic method and singularity analysis. We provide a detailed explanation of the quadratic method in Section 3.9.2, tailored to the problem in this thesis. In this

34

2.3. MAP ENUMERATION AND SINGULARITY ANALYSIS

section, we review some definitions related to singularity analysis. A few important results about maps derived using this technique are presented in Sections 3.3 and 3.4. 2.3.1. Analytic Functions and Singularities. All functions in this subsection are complex-valued. A generating function F pzq with complex-valued coefficients pFn : n P Ně0 q is a ř formal power series F pzq “ nPNě0 Fn z n . Let Ω be an open connected subset of the complex plane. A function F defined over Ω is analytic at a point z0 P Ω if, for z in some open disc centred at z0 and contained in Ω, it is representable by a convergent power series expansion F pzq “ ř n nPNě0 Fn pz ´ z0 q . A function is analytic in a region Ω if and only if it is analytic at every point of Ω. If F is analytic at a point z0 , then there exists a disc such that the series representing F pzq is convergent for z inside the disc and divergent for z outside the disc. The disc is called the disc of convergence and its radius is the radius of convergence of F at z0 . A singularity of F is a point r where F ceases to be analytic; we say that F is singular at r. If F is analytic at 0, we define the dominant singularities of F to be the singularities on the boundary of the disc of convergence. Pringsheim’s Theorem tells us that if F is representable at 0 by a series expansion that has non-negative coefficients and radius of convergence rF , then the point rF is a singularity of F ; see [36, Theorem IV.6]. The singular value of F is F prF q, which may be infinite. Given a complex number z, we write |z| for its modulus and argpzq for its argument. For a complex number r ‰ 0, a function is ∆-analytic at r if it can be analytically continued in the domain ˇz ˇ ˇ ´z ¯ˇ ) ˇ ˇ ˇ ˇ ∆pφ, η, rq :“ z P C : ˇ ˇ ă 1 ` η, ˇarg ´1 ˇąφ , r r !

for some real numbers φ, η with η ą 0 and 0 ă φ ă π2 . A function F is said to admit a singular expansion at r if it is ∆-analytic at r and J ÿ

ˆˇ ´ z ¯αn ˇ ` O ˇ1 ´ F pzq “ Fn 1 ´ r n“0

z ˇˇA ˇ r

˙

35

2.3. MAP ENUMERATION AND SINGULARITY ANALYSIS

uniformly in z P ∆pφ, η, rq, for a sequence of complex numbers pFn : 0 ď n ď Jq and an increasing sequence of real numbers pαn : 0 ď n ď Jq satisfying αn ă A for all 0 ď n ď J and αn P Ně0 for at least some n. Alternative definitions appear in [36, Definition VI.1] and [35, Definition 1]. 2.3.2. The Airy Distribution. Banderier, Flajolet, Schaeffer & Soria [16] established the common occurrence of a probability distribution, the Airy distribution, which quantifies the sizes of highly connected blocks in many different maps. In this subsection, we overview basic definitions, deferring more details to Sections 3.3 and 3.4. The Airy distribution is the probability distribution whose density is Apxq “2e´2x

3 {3

`

xAipx2 q ´ Ai1 px2 q

˘

˙ ˆ 2n 1 ÿ 2nπ 2{3 n Γp 3 ` 1q “ , p´x3 q sin ´ πx nPN n! 3 where the Airy function Ai is given by 1 Aipzq “ 2π

ż8

t3

eipzt` 3 q dt ´8

ˆ ˙ n`1 ÿ 2pn ` 1qπ 1 1{3 n Γp 3 q sin . “ p3 zq π32{3 nPN n! 3 ě0

36

Joint Convergence of Random Quadrangulations and Their Cores

38

CHAPTER 3

Joint Convergence of Random Quadrangulations and Their Cores 3.1. Introduction Much work has been devoted to understanding the asymptotic properties of large random planar maps. It is conjectured and known in several cases, that after rescaling the graph distance properly, planar maps from many families converge to the same universal metric space, the Brownian map, in the Gromov-HausdorffProkhorov sense. Recently Le Gall [48] and Miermont [59] independently proved that the Brownian map is the scaling limit of several important families of planar maps, and Addario-Berry & Albenque [4] proved that simple triangulations and simple quadrangulations also rescale to the same limit object. Further results, the aim of this chapter is to show that random quadrangulations and their cores jointly converge to the same limit object, even after conditioning on their sizes. Before making this more precise, we state one corollary (Theorem 3.1.1) of our main result: the Brownian map is again the scaling limit of random 2-connected quadrangulations. Recall that a graph is 2-connected if the removal of any vertex does not disconnect the graph. A simple graph is a graph without multiple edges and loops. We write Q, R, and S for the set of rooted connected, 2-connected, and simple quadrangulations, respectively. It is easy to verify that simple quadrangulations are 2-connected, so S Ă R Ă Q. It is technically convenient to view a single edge as a 2-connected, simple quadrangulation, and we do this.

3.1. INTRODUCTION

39

Given a graph G, we write dG for the graph distance on vpGq. If G is finite write µG for the uniform probability measure on vpGq. For c ą 0, write cG “ pvpGq, c ¨ dG , µG q for the measured metric space. Recall that we write m8 for the Brownian map, and that Rr “ tR P R : |vpRq| “ ru. Theorem 3.1.1 Let Rr Pu Rr , then as r Ñ 8, ˆ

21 40r

˙1{4 Rr Ñ m8

in distribution for the Gromov-Hausdorff-Prokhorov topology.

A brief overview of the Gromov-Hausdorff-Prokhorov (GHP) distance appears in Section 3.2.2. Fix a bipartite map M “ pM, uvq. A cycle C in a map M is nearly facial if at least one connected component of S2 zC contains no vertices of M (it may contain edges). We say M is nearly simple if every cycle in M with length two is nearly facial. Write M˝ “ pM ˝ , uvq for the map obtained by collapsing each nearly facial 2-cycle into an edge. (This is a slight abuse of notation as the edge uv P epM q may be collapsed with other edges in forming M ˝ , but the meaning should be clear.) Note that M is nearly simple precisely if M˝ is simple – in this case we call M˝ the simple nerve of M. For A Ă vpM q, write M rAs for the submap of M induced by A. For any edge e P epM q with endpoints x and y let Be Ă vpM q be maximal subject to the constraints that tx, yu Ă Be , and that M rBe s is 2-connected. We call M rBe s˝ a 2-connected block of M. In particular, write R‚ “ R‚ pMq “ pM rBuv s˝ , uvq and call R‚ the 2connected root block of M. Our choice to collapse nearly-facial 2-cycles renders this different from the standard graph theoretic definition of a 2-connected block. We make this choice as it simplifies upcoming counting arguments.

3.1. INTRODUCTION

40

a

b g

c

h i

d e f

Figure 3.1. The 2-connected blocks of M are M rta, b, c, dus˝ and M rtd, e, f, g, h, ius˝ . The simple blocks of M are M rta, b, c, d, e, f, ius˝ and M rtd, g, h, ius˝ .

Next, for any edge e P epM q, consider the set S “ tB Ă vpM q : tx, yu Ă B, M rBs is nearly simpleu. Let S 1 “ tB 1 P S : B 1 is maximalu, where maximal is with respect to the inclusion relation on vpM q. Then define Be1 Ă vpM q to be the lexicographically minimal element of S with respect to the total order ăM . We call 1 ˝ s , uvq and call S‚ the M rBe1 s˝ a simple block of M. Write S‚ “ S‚ pMq “ pM rBuv

simple root block of M. Recall from Section 2.1.4 that ăM is a total order on the edge set of M. Write RpMq (resp. SpMq) for the largest 2-connected (resp. simple) block of M, rooted at its ăM -minimal edge, and write bpMq “ |vpRpMqq| and sbpMq “ |vpSpMqq|. If there are multiple 2-connected blocks with size bpMq, among these blocks we take RpMq to be the one whose root edge is ăM -minimal, and use the same convention for SpMq. We call RpMq and SpMq the 2-connected and simple cores of M, respectively. The next theorem states that a uniform quadrangulation, its largest 2-connected block, and its largest simple block jointly converge to the same Brownian map. (Note that the definition of Rq in the coming theorem is different from that in Theorem 3.1.1. We recycle some notation to keep the sub- and superscripts from becoming too cumbersome; we will always remind the reader when there is a possibility of ambiguity or confusion.)

3.1. INTRODUCTION

41

Theorem 3.1.2 Let Qq Pu Qq and write Rq “ RpQq q, Sq “ SpQq q. Then as q Ñ 8, ˜ˆ

9 8q

˙1{4

ˆ Qq ,

9 8q

˙1{4

ˆ Rq ,

9 8q

¸

˙1{4 Sq

Ñ pm8 , m8 , m8 q

in distribution for the Gromov-Hausdorff-Prokhorov topology.

The convergence of the first coordinate in Theorem 3.1.2 was proved independently by Le Gall [48, Theorem 1.1] (Theorem 1.1.1) and by Miermont [59, Theorem 1] (Theorem 1.1.2). The convergence of the third coordinate on its own is implied by a result by Addario-Berry & Albenque [4, Theorem 1.1] (Theorem 1.3.1), who show d

that if Sq is a uniform simple quadrangulation for all q, then p3{p8|vpSq q|qq1{4 Sq Ñ m8 . It is known [16, 37] that |vpSq q|{q Ñ 1{3 in probability, so in the third coordinate the scaling factor p9{p8qqq1{4 may be replaced by p3{p8|vpSq q|qq1{4 , and the convergence then follows from the result of [4]. Similarly, the convergence of the second coordinate on its own can be deduced from Theorem 3.1.1. Theorem 3.1.2 and Theorem 3.1.1 follow from a stronger “local invariance principle”, in which the sizes of the largest 2-connected block and largest simple block are fixed rather than random. Given integers q ě r ě s ě 1, let Qq,r,s “ tQ P Qq : bpQq “ r, sbpQq “ su , Rr,s “ tQ P Rr : sbpQq “ su .

3.1. INTRODUCTION

42

Theorem 3.1.3 ` ˘ Let prpqq : q ě 1q and pspqq : q ě 1q be such that rpqq “ 7q{15 ` O q 2{3 and spqq “ q{3 ` Opq 2{3 q as q Ñ 8. Let Qq Pu Qq,rpqq,spqq and write Rq “ RpQq q, Sq “ SpQq q. Then as q Ñ 8, ˜ˆ

9 8q

˙1{4

ˆ Qq ,

9 8q

˙1{4

ˆ Rq ,

9 8q

¸

˙1{4 Sq

Ñ pm8 , m8 , m8 q

in distribution for the Gromov-Hausdorff-Prokhorov topology.

We provide an outline of the proof of Theorem 3.1.3 (our main result) in Section 3.1.2. Now and for the remainder of the chapter, fix C ą 0 and let prpqq : q ě 1q and pspqq : q ě 1q be such that |rpqq ´ 7q{15| ă Cq 2{3 and |spqq ´ 5q{7| ă Cq 2{3 for all q sufficiently large. The scaling of rpqq and spqq in Theorem 3.1.3 is explained by the following local limit theorem for the asymptotics of maximal block sizes. Theorem 3.1.4 Let Qq Pu Qq , and write δr pqq “

rpqq´7q{15 , q 2{3

P pbpQq q “ rpqq, sbpQq q “ sprpqqqq “

δs pqq “

sprpqqq´5rpqq{7 . rpqq2{3

Then

βA pβδs pqqq β 1 A pβ 1 δr pqqq p1 ` op1qq , rpqq2{3 q 2{3

where β and β 1 are positive constants given in Propositions 3.4.2 and 3.4.1 respectively, A : R Ñ r0, 1s is the Airy density.

Here op1q denotes a function tending to zero whose decay may depend on C, but we omit this dependence from the notation. We prove Theorem 3.1.4 using the machinery developed by Banderier, Flajolet, Schaeffer & Soria [16], based on singularity analysis of generating functions, in Section 3.4. Theorem 3.1.2 follows from Theorem 3.1.3, Theorem 3.1.4, and an easy averaging argument. We similarly deduce Theorem 3.1.1 by averaging over the second coordinate in the next proposition.

3.1. INTRODUCTION

43

Proposition 3.1.1. Let Rr Pu Rr,sprq and write Sr “ SpRr q. Then as r Ñ 8, ˜ˆ

21 40r

˙1{4

ˆ Rr ,

21 40r

¸

˙1{4 Sr

Ñ pm8 , m8 q

in distribution for the Gromov-Hausdorff-Prokhorov topology.

Remarks. (1) The proof of Proposition 3.1.1, given in Section 3.7, uses the convergence of simple quadrangulations, proved in [4], to deduce convergence of 2connected quadrangulations, as a stepping stone to proving the joint convergence of Theorem 3.1.3. The results of [4] in turn use the “rerooting invariance trick” introduced by Le Gall [48], together with the convergence of uniform quadrangulations to the Brownian map [48, 59], to deduce convergence for uniform simple quadrangulations. We mention this to emphasize that the results of this chapter do not constitute an independent proof of convergence for uniform quadrangulations. (2) In [4] it is also shown that simple triangulations converge to the Brownian map. Using this, the arguments of the current chapter could be modified to show joint convergence of uniformly random triangulations and their largest loopless and simple blocks. Before sketching our proof, we first describe the combinatorial relations between Q, R‚ pQq and S‚ pQq, on which our proofs rely. 3.1.1. Bijections for Q, R and S. Suppose we are given only R‚ “ R‚ pQq. What additional information is required to reconstruct Q? Similarly, what do we require in addition to S‚ “ S‚ pR‚ q in order to reconstruct R‚ ? In each case, the reconstruction requires augmenting the edges with additional data. The reconstruction (equivalently described as decomposition) procedures, which we describe in this section, are all either due to Tutte [65] or are obtained by slight variants of his methods.

3.1. INTRODUCTION

When reconstructing R‚ from S‚ , this data consists of a 2-connected quadrangulation for each edge of S‚ . When reconstructing Q from R‚ , we require a sequence of quadrangulations for each edge of R‚ , together with a second, binary sequence whose entries specify how to attach the quadrangulations in the sequence. In both cases, the root edge must be treated slightly differently from the others (in brief, for the root edge we must specify data twice, once for each side of the edge). We now turn to details. A quadrangulation of a 2-gon is a rooted map whose unbounded face has degree 2, with all other faces of degree 4, rooted such that the unbounded face lies to the left of the root edge. Temporarily write T for the set of quadrangulations of 2-gons. Given a map in T , merge the two edges incident to the unbounded face to obtain a map in Q; we call this the natural bijection between T and Q. For n ě 3, it in fact restricts to a bijection between Tn and Qn . Also, T2 contains only one element: the map with one edge and two vertices. Recalling that we also view a single edge as a 2-connected quadrangulation, it follows that T2 “ Q2 , and it is convenient to view the natural bijection as associating these two sets with one another. Let S “ pS, uvq be a simple quadrangulation. List the vertices of S in breadthfirst order as u1 , . . . , un and list the edges of S as uv “ e1 , . . . , em , oriented so that the tail precedes the head in breadth-first order. To build a 2-connected quadrangulation with simple root block S, proceed as follows (see Figure 3.2).

(1) Create a second copy e0 of the edge uv so that e0 lies to the left of e1 . (2) For 0 ď i ď m let Mi be a 2-connected quadrangulation, and let M1i “ pMi , ui vi q be the quadrangulation of a 2-gon associated to Mi by the natural bijection. (3) For each 0 ď i ď m, identify the edge ei with the root edge ui vi of M1i . The resulting map has a single facial 2-cycle (lying between M0 and M1 ), with vertices u and v; collapse it and root at the resulting edge uv.

44

3.1. INTRODUCTION

45

e3 e1

e4 M0

e2

M1

(a)

M2 M3

M4

(b)

(c)

(d)

Figure 3.2. (a) A simple quadrangulation. (b) “Decorations” for the edges. (c) After attaching the decorations. (d) The map R.

Call the resulting map R. Then R is a 2-connected quadrangulation with S‚ pRq “ S. We note that (3.1.1) |epSq| ÿ

|epRq| “ |epSq| ` i“0

|epMi q| 1r|epMi q|‰1s “ ´1 `

|epSq| ÿ

p1 ` |epMi q| 1r|epMi q|‰1s q .

i“0

Proposition 3.1.2. The above procedure induces a bijection ϕ between R and the set tpS, Θq : S P S, Θ P R|epSq|`1 u . Proof. Given a 2-connected quadrangulation of a 2-gon, collapsing the unbounded face to form a single edge (which is equivalent to taking the simple nerve), then rooting at this edge, yields a 2-connected quadrangulation. This operation is easily seen to be a bijection. In view of the fact that the quadrangulation R P R in the above construction has S‚ pRq “ S, the result follows.



Next let R “ pR, uvq be a 2-connected quadrangulation and list epRq as e1 , . . . , em , ´ as above. For each 1 ď i ď m, write e` i and ei for the head and the tail of ei re-

spectively. To build a quadrangulation with 2-connected root block R, proceed as follows (see Figure 3.3). (1) Create a second copy e0 of the edge uv so that e0 lies to the left of e1 .

3.1. INTRODUCTION

46

e2

e1

e3

M0,1

e4

(a)

M1,1

M4,1

M4,2

(b)

e2,1 e0,1 e0,2 e1,1

e4,1 e3,1

e1,2

e4,2 e4,3

(c)

(d)

Figure 3.3. The quadrangulation in (d) can be reconstructed from its 2-connected core in (a) with the decoration ppLi , bi q : 0 ď i ď rq where L0 “ pM0,1 q, b0 “ p1q, L1 “ pM1,1 q, b1 “ p0q, L2 “ L3 “ H, b2 “ b3 “ H, L4 “ pM4,1 , M4,2 q, b4 “ p0, 1q.

(2) For 0 ď i ď m fix `i P Ně0 and sequences Li “ pMi,j , 1 ď j ď `i q P Q`i , bi “ pbi,j , 1 ď j ď `i q P t0, 1u`i . (3) For each 1 ď i ď m, add an additional `i copies of ei ; label the resulting `i ` 1 copies of ei as ei,1 , . . . , ei,`i `1 in clockwise order around e´ i . (4) For 0 ď i ď m and 1 ď j ď `i , let M1i,j be the quadrangulation of a 2-gon associated to Mi,j by the natural bijection. (5) Attach M1i,j “ pMi,j , ui,j vi,j q inside the 2-cycle formed by ei,j and ei,j`1 by ` identifying ui,j with e´ i (if bi,j “ 0) or ei (if bi,j “ 1). The resulting map

has a single facial 2-cycle, with edges e0,`0 `1 and e1,1 ; collapse it and root at the resulting edge uv. Call the resulting map Q. Then Q is a connected quadrangulation with R‚ pQq “ R. We note that |epRq| `i ÿ ÿ

|epQq| “ |epRq| `

p|epMi,j q| ` 1 ` 1r|epMi,j q|‰1s qq

i“0 j“1

3.1. INTRODUCTION

(3.1.2)

47

|epRq| ÿ ˆ

“ ´1 ` i“0

1`

`i ÿ

˙ p|epMi,j q| ` 1 ` 1r|epMi,j q|‰1s q .

j“1

In the following proposition we write pQ ˆ t0, 1uq˚ “ tHu Y

Ť

ně1 pQ

ˆ t0, 1uqn .

Proposition 3.1.3. The above procedure induces a bijection ψ between Q and the set ! ) ` ˘ ˚ |epRq|`1 pR, Γq : R P R, Γ P pQ ˆ t0, 1uq . Proof. This is immediate from the fact that the above construction has R‚ pQq “ R.

 For both decompositions, we refer informally to the maps in the vectors Θ and

Γ as decorations or as pendant submaps.

3.1.2. Proof sketch for Theorem 3.1.3. In this subsection, we assume familiarity with the Gromov-Hausdorff and Gromov-Hausdorff-Prokhorov distances. The relevant definitions appear in Section 3.2. We begin by stating (and sketching the proof of) a joint convergence result for a 2-connected quadrangulation and its largest simple block; the proof of this result contains most of the key ideas for the proof of Theorem 3.1.3. Given Rr “ pRr , er q Pu Rr,sprq , it is easily seen that Sr “ SpRr q is uniformly d

distributed over Ssprq . Then by [4, Theorem 1], p3{8sprqq1{4 Sr Ñ m8 as sprq Ñ 8. 3 21 ´1{4 Also, the definition of sprq guarantees that p 8sprq q1{4 ¨ p 40r q Ñ 1 as r Ñ 8.

Let e1 be the ăRr -minimal oriented edge of Sr . If er P epSr q then Sr “ S‚ pRr q. Write R1r “ pRr , e1 q. By Proposition 3.1.2, R1r uniquely decomposes as ϕpR1r q “ pS, Θq P Ssprq ˆ R|epSq|`1 , and our choice of e1 guarantees that S “ Sr . Write Θ “ pΘi , 0 ď i ď 2sprq ´ 4q, and LpRr q “ max p|vpΘi q| : 0 ď i ď 2sprq ´ 4q , DpRr q “ max pdiampΘi q : 0 ď i ď 2sprq ´ 4q .

3.1. INTRODUCTION

In words, LpRr q and DpRr q are the greatest number of vertices and the greatest diameter, respectively, of any submap pendant to the biggest simple block of Rr . The identification of Sr as a submap of Rr gives the bound dGH pRr , Sr q ď DpRr q. ` 21 ˘1{4 To prove that 40r dGH pRr , Sr q “ op1q in probability, it thus suffices to show that ` 21 ˘1{4 DpRr q “ op1q in probability. (Note that here we have the Gromov-Hausdorff 40r rather than Gromov-Hausdorff-Prokhorov distance!) To accomplish this, we use the methodology developed by Banderier, Flajolet, Schaeffer & Soria [16], which allows one to describe the largest block size of a map whenever the map may be described by a recursive decomposition into rooted blocks, using a suitable composition schema; this is explained in greater detail in Section 3.4. We thereby obtain the following distributional result for |vpSr q|. Proposition 3.1.4. Let Rr Pu Rr , then for any A ą 0, uniformly over x P r´A, As, ` ˘ βA pβxq P sbpRr q “ t5r{7 ` xr2{3 u “ p1 ` op1qq , r2{3 where β is given in Proposition 3.4.2. The proof of Proposition 3.1.4 appears in Section 3.4. The range of values for r in the above local limit theorem is what leads to our choice for the range of sprq in Theorem 3.1.3 and Theorem 3.1.4. The following proposition bounds the size of the largest simple block of a random 2-connected quadrangulation. Proposition 3.1.5. For any A ą 0, there exist positive constants c1 and c2 such ` ‰ that for all r P N and for integer k P 5r{7 ` Ar2{3 , r , if Rr Pu Rr , ` ˘ P psbpRr q “ kq ď c1 exp ´c2 r pk{r ´ 5{7q3 . This proposition is a slight extension of [16, Theorem 1], which proves similar bounds but requires that pr ´ kq{r2{3 Ñ 8. We do not reprove the entire result, but simply analyze the behaviour in the range not covered in the work of [16]. We use Proposition 3.1.5 in proving stretched exponential tail bounds for the size of the largest pendant submap in a random 2-connected quadrangulation.

48

3.1. INTRODUCTION

Proposition 3.1.6. For all ε P p0, 1{3q, there exist positive constants c1 , c2 , and c3 “ c3 pεq such that for all r P N, if Rr Pu Rr,sprq , ` ˘ P LpRr q ě r2{3`ε ď c1 exp p´c2 rc3 q . Proofs for Proposition 3.1.5 and Proposition 3.1.6 are given in Section 3.5. Next we deduce a bound for DpRr q through extending a result by Chassaing & Schaeffer [28]. The following proposition follows straightforwardly from [28, Proposition 4]. Proposition 3.1.7. ([28]). There exist positive constants y0 , C1 , and C2 such that for all y ą y0 and q P N, if Qq Pu Qq , ` ˘ P diam pQq q ą yq 1{4 ď C1 expp´C2 yq . This bound is for connected quadrangulations rather than 2-connected ones. However, at the cost of polynomial corrections, we are able to transfer the result to the latter family of quadrangulations, as shown in Section 3.5.1. This in particular yields the following bound. Proposition 3.1.8. Let Rr Pu Rr,sprq , then there exist positive constants C1 , C2 , and C3 such that ` ˘ ` ˘ P DpRr q ě r5{24 ď C1 exp ´C2 rC3 . The above results immediately give rise to Gromov-Hausdorff convergence of pRr , Sr q after rescaling, as shown in Proposition 3.5.2 in the end of Section 3.5.1. However, to deduce GHP convergence, the above propositions are insufficient, as they do not guarantee that the uniform measures on vpRr q and vpSr q are close in the Prokhorov sense. Here is an example of the sort of issue that may a priori still occur. For all s P N, let Ss Pu Ss have root edge es . Let Ps be the quadrangulation of a 2-gon with 2ts{5u`2 vertices composed of parallel alternating 1-paths and 3-paths, and write e1s for one of the boundary edges of Ps . Then identify es with e1s , embed Ps

49

3.1. INTRODUCTION

50

es

Figure 3.4. Parallel alternating 1-paths and 3-paths attached to the root edge es .

in the face of Ss to the left of es , and write R1s for the resulting quadrangulation; see Figure 3.4. Recall that m8 “ pM, D1˚ , µ1 q is the Brownian map. Then it is not hard to see that pR1s , Ss q converges after rescaling to pm18 , m8 q, where m18 “ pM, D1˚ , µ11 q has the geometry of the Brownian map but has mass measure 57 µ1 ` 27 δρ , where ρ is a point of M with law µ1 . To prevent the masses of “pendant submaps” from concentrating on small regions in this manner, we use that they attach to exchangeable random locations on the simple block and that each of them has asymptotically negligible size. The first follows from the details of the construction of a 2-connected quadrangulation from its simple root block, explained in Section 3.1.1; the second is a consequence of Proposition 3.1.6. In order to show that the facts from the preceding paragraph suffice to imply joint convergence, we prove a general result on the preservation of Gromov-HausdorffProhkorov convergence under small random modifications; our result relies on results of Aldous on concentration for partial sums of exchangeable random variables. Details for this part of the proof appear in Sections 3.6 and 3.7. We conclude the proof sketch by explaining how we strengthen Proposition 3.1.1 to prove Theorem 3.1.3. First, with Qq Pu Qq,rpqq,spqq , we show that RpQq q contains SpQq q with high probability. The joint convergence of the second and third coordinates in Theorem 3.1.3 then follows from Proposition 3.1.1.

3.2. PRELIMINARIES

51

The convergence of the first coordinate does not follow from the existing result by Le Gall [48] or Miermont [59], because Qq here is not uniformly distributed over Qq , but conditioned on bpQq q “ rpqq and sbpQq q “ spqq. To deal with this, we require versions of Propositions 3.1.6 and 3.1.8 that apply to uniform quadrangulations sampled from Qq and Qq,rpqq,sprpqqq . These follows straightforwardly once we show that with high probability, SpRpQq qq “ SpQq q. We postpone the details. A reprise of the argument for Proposition 3.1.1 then shows that if Qq Pu Qq,rpqq,spqq , d

d

then pQq , RpQq qq Ñ pm8 , m8 q as q Ñ 8. Since we also know pRpQq q, SpQq qq Ñ pm8 , m8 q as q Ñ 8, Theorem 3.1.3 follows immediately.

3.2. Preliminaries 3.2.1. Hausdorff and Prokhorov distances. Let pV, dq be a compact metric space with its Borel σ-algebra BpV q. Given A Ă V , the ε-neighborhood of A is defined as Aε “ Aεd “ tx P V : Dy P V, dpx, yq ă εu . The Hausdorff distance dH between two non-empty subsets X, Y of pV, dq is defined as dH pX, Y q “ inf pε ą 0 : X Ă Y ε , Y Ă X ε q .

Denote by PpV q the collection of all probability measures on the measurable space pV, BpV qq. The L´evy-Prokhorov distance dP : PpV q2 Ñ r0, 8q between two Borel probability measures µ and ν on V is given by dP pµ, νq “ inf pε ą 0 : µpAq ď νpAε q ` ε and νpAq ď µpAε q ` ε, @A P BpV qq .

3.2.2. Gromov-Hausdorff(-Prokhorov) distance. We refer the reader to [26] and [48, 59] for more details on the Gromov-Hausdorff and Gromov-HausdorffProkhorov distances and the topologies they induce. Let pV, dq and pV 1 , d1 q be two

3.3. COMPOSITION SCHEMATA

compact measured metric spaces. A correspondence between V and V 1 is a set C Ă V ˆ V 1 such that for every x P V , there is x1 P V 1 with px, x1 q P C, and vice versa. We write CpV, V 1 q for the set of correspondences between V and V 1 . The distortion of any set C Ă V ˆ V 1 with respect to d and d1 is given by dis pC; d, d1 q “ sup p|dpx, yq ´ d1 px1 , y 1 q| : px, x1 q P C, py, y 1 q P Cq . The Gromov-Hausdorff distance between pV, dq and pV 1 , d1 q is defined as dGH ppV, dq, pV 1 , d1 qq “ inf pε ą 0 : DC P CpV, V 1 q, dispC; d, d1 q ď 2εq .

Next, suppose µ and µ1 are non-negative finite Borel measures on pV, dq and pV 1 , d1 q, respectively. A measured metric space is a quadruple pV, d, µq where pV, dq is a metric space and µ is a non-negative finite Borel measure on V . The GromovHausdorff-Prokhorov distance between V “ pV, d, µq and V1 “ pV 1 , d1 , µ1 q is given by dGHP pV, V1 q “ inf pmax tδH pφpV q, φ1 pV 1 qq, δP pφ˚ µ, φ1˚ µ1 quq where the infimum is taken over all isometries φ, φ1 from pV, dq, pV 1 , d1 q into a metric space pZ, δq (see Miermont [58, Section 6.2]). Writing K for the set of all isometry classes of compact metric spaces, pK, dGHP q is a Polish space; when we refer to GHP convergence we mean convergence in this space. 3.3. Composition Schemata Let F pzq “

ř ně0

Fn z n be a generating function (i.e. an analytic function with

nonnegative integer coefficients) with positive radius of convergence r “ rF . We say F is singular with exponent 3{2 if the following properties hold. ‚ There exists ε ą 0 such that F is continuable in ∆ “ tz : |z| ă r ` ε, z R rr, r ` εqu. ‚ There exist positive constants a “ aF , b “ bF , c “ cF such that F pzq “ a ´ bp1 ´ z{rq ` cp1 ´ z{rq3{2 ` Opp1 ´ z{rq2 q as z Ñ r in ∆.

52

3.3. COMPOSITION SCHEMATA

53

Gao and Wormald [37] derived asymptotics for the coefficients of F under the above conditions. Proposition 3.3.1 ([37], Theorem 1 (iii)). Let F be singular with exponent 3{2, let r and c be as above. Then Fn „

3c r´n . 4π 1{2 n5{2

Next, let C and H be generating functions with positive coefficients, and define a bivariate generating function M by M pz, uq “ CpuHpzqq; Banderier, Flajolet, Schaeffer & Soria [16] call this a composition schema. We generically write Ck “ rz k sCpzq and Mn “ rz n sM pz, 1q, and for n ě 1 let Xn be a real random variable with law given by P pXn “ kq “

Ck n rz sHpzqk . Mn

We quote from [16]: Combinatorially, this corresponds to a composition M “ C ˝ H between classes of [rooted] objects, where objects of type H are substituted freely at individual “atoms” (e.e., nodes, edges, or faces) of elements of C... rz n uk sM pz, uq gives the number of M-ojbects of total size n whose C-component (the “core”) has size k, and Xn is the corresponding random variable describing core-size in this general context. More precisely, Xn is the law of the size of the C-component containing the root, in an object chosen uniformly at random from among all M-objects of size n. The connection with the bijections for quadrangulations described in Section 3.1.1 should be clear. We say the triple pM, C, Hq is a map schema if C and H are both singular with exponent 3{2 and additionally HprH q “ rC .1 The latter condition heuristically states that the singularity of M pzq “ M pz, 1q closest to the origin is caused by 1In

[16], this is called a critical composition schema of singular type p 23 ˝ 32 q. We shorten this to “map schema” as such schemata seem to primarily arise in the study of maps.

3.3. COMPOSITION SCHEMATA

54

the singularity of C, rather than by a failure of the implicit function theorem/the Lagrange inversion theorem. The following results are all from [16]. Proposition 3.3.2 ([16], Theorems 1 and 5). Suppose pM, C, Hq is a map schema with ` ˘ Cpzq “ c0 ´ c1 p1 ´ z{rC q ` c3{2 p1 ´ z{rC q3{2 ` O p1 ´ z{rC q2 , ` ˘ Hpzq “ h0 ´ h1 p1 ´ z{rH q ` h3{2 p1 ´ z{rH q3{2 ` O p1 ´ z{rH q2 , the expansions for Cpzq and Hpzq valid in some neighbourhoods of rC and of rH , respectively. Let α “ αpM,C,Hq , β “ βpM,C,Hq and γ “ γpM,C,Hq be defined by 1{2

α“

3{2

c1 h3{2 h0 ` c3{2 h1 3{2

h0

3{2

5{3 c3{2 h1 h1 ,β “ . ,γ “ β ¨ 3{2 2{3 p3h3{2 q h0 α ¨ h0

Then rz n sM pz, 1q „

´n 3α rH . 4π 1{2 n5{2

Furthermore, writing α0 “ α0,pM,C,Hq “ h0 {h1 , for any A ą 0, lim

` ˘ sup |n2{3 P Xn “ tα0 n ` xn2{3 u ´ γApβxq| “ 0 .

nÑ8 xPr´A,As

Finally, there exist continuous functions f : rα0 , 1s Ñ p0, 8q and g : rα0 , 1s Ñ p0, 8q such that for any function λ : R Ñ R with λpnq Ñ 8 and λpnq “ opn1{3 q, P pXn “ kq “ p1 ` op1qqf pk{nq

pk{n ´ α0 q1{2 ´npk{n´α0 q3 gpk{nq e , n1{2 p1 ´ k{nq3{2

the preceding asymptotic holding uniformly in α0 n ` n2{3 λpnq ă k ă n ´ n2{3 λpnq. Remark. In [16], schema of the form M pz, uq “ CpuHpzqq ` Dpzq are also considered. Replacing M pz, uq by M pz, uq ´ Dpzq turns this into a compositional schema; if the latter is a map schema then Proposition 3.3.2 applies. The equation involving D is convenient when considering map families in which the core may have size zero; such families should be counted by ru0 sM pz, uq, which is identically zero in CpuHpzqq.

3.3. COMPOSITION SCHEMATA

55

Corollary 3.3.3 ([16], Theorem 7). Suppose pM, C, Hq is a map schema with α0 “ α0,pM,C,Hq and β “ βpM,C,Hq defined in Proposition 3.3.2. Let Xn˚ be the size of the largest C-component in a random M-map of size n with uniform distribution. Then (3.3.1)

` X \˘ βApβxq P Xn˚ “ α0 n ` xn2{3 “ p1 ` op1qq , n2{3

uniformly for x in any bounded interval. Furthermore, there exist continuous functions f : rα0 , 1s Ñ p0, 8q and g : rα0 , 1s Ñ p0, 8q such that for any function λ : R Ñ R with λpnq Ñ 8 and λpnq “ opn1{3 q, (3.3.2)

P pXn˚ “ kq “ p1 ` op1qqf pk{nq

pk{n ´ α0 q1{2 ´npk{n´α0 q3 gpk{nq , e n1{2 p1 ´ k{nq3{2

the preceding asymptotic holding uniformly in α0 n ` n2{3 λpnq ă k ă n ´ n2{3 λpnq.

Corollary 3.3.4. Suppose pM, C, Hq is a map schema. Let α0 “ α0,pM,C,Hq and g be as in Proposition 3.3.2, and let Xn˚ be defined as in Corollary 3.3.3. Then for any function λ : R Ñ R with λpnq Ñ 8 and λpnq “ opn1{3 q, and for any a ą 0, P pXn˚ “ kq “ Θp1q ¨

pk{n ´ α0 q1{2 ´npk{n´α0 q3 gpk{nq e , n1{2 p1 ´ k{nq3{2

uniformly over integers k P rα0 n ` an2{3 , n ´ λpnqn2{3 q.

Proof. Note that if k “ α0 n ` cn2{3 then pk{n ´ α0 q1{2 ´npk{n´α0 q3 gpk{nq c1{2 3 1{3 e “ e´c gpα0 `c{n q . 1{2 3{2 2{3 3{2 n p1 ´ k{nq n p1 ´ k{nq For |k´α0 n| “ Opn2{3 q, the latter is Θpn´2{3 q. By (3.3.1), we indeed have P pXn˚ “ kq “ Θpn´2{3 q for such k. If the claim of the corollary fails then there exists a sequence pni , i ě 1q and ` ˘ 2{3 ki P rα0 ni ` an2{3 , ni ´ λpni qni s along which the ratio of P Xn˚i “ ki and pki {ni ´ α0 q1{2 1{2 ni p1

´ ki {ni

q3{2

3 gpk

e´ni pki {ni ´α0 q

i {ni q

3.3. COMPOSITION SCHEMATA

56

either diverges or tends to zero. By passing to a subsequence if necessary, we may 2{3

assume that either ki ´ α0 ni “ Opn2{3 q or pki ´ α0 ni q{ni

Ñ 8. In view of the

above computation, the first possibility is in contradiction with (3.3.1). The second possibility is in contradiction with (3.3.2); thus neither can occur.



Let M pzq, Cpzq, Bpzq be the generating functions of rooted connected, 2-connected, and simple quadrangulations respectively. More precisely, we take rz n sM pzq “ |Qn`2 |, rz n sCpzq “ |Rn`2 |, and rz n sBpzq “ |Sn`2 | for n ě 1, and rz n sM pzq “ rz n sCpzq “ rz n sBpzq “ 0 for n “ 0. (The latter is slightly at odds with our convention of viewing a single edge as a 2-connected quadrangulation, but is algebraically convienent below.) Define ˆ (3.3.3)

(3.3.4)

Hpzq “ z

1 1 ´ 2zp1 ` M pzqq

˙2 ,

U pzq “ z p1 ` Cpzqq2 .

The following two lemmas follow immediately from Propositions 3.1.2 and 3.1.3 respectively. Lemma 3.3.5. We have the following substitution relation between M pzq and Cpzq: (3.3.5)

M pzq “ CpHpzqq ¨

2zp1 ` M pzqq 1 ` . 1 ´ 2zp1 ` M pzqq 1 ´ 2zp1 ` M pzqq

Equivalently, (3.3.6)

M pzq “ CpHpzqq ` 2zp1 ` M pzqq2 .

(3.3.6) is obtained by multiplying both sides of (3.3.5) by 1 ´ 2zp1 ` M pzqq and then rearranging elements. To see that (3.3.6) gives a composition schema, we can ˆ pzq “ CpHpzqq with M ˆ pzq “ M pzq ´ 2zp1 ` M pzqq2 . rewrite it as M We now take a closer look at equation (3.3.3), which describes the “M -decorations” of an edge of a C-object (i.e. of a 2-connected map). This is best understood with the

3.4. AIRY DISTRIBUTION FOR QUADRANGULATIONS

bijection from Proposition 3.1.2 at hand. In the term 2zp1 ` M pzqq, the multiplier 2 counts the choice of extremity at which the decoration is attached; M pzq counts the case when attachment is a quadrangulation with at least 3 vertices (recalling that z marks the number of vertices less two, and the lowest power term of M pzq is 2z); the additive term 1 counts the case when attachment is a single edge; the multiplier z adjusts the counting of extra vertices resulting from the attachment (we multiply by z instead of z 2 because the attachment vertex is already counted in the core). Taking the reciprocal of 1 ´ 2zp1 ` M pzqq accounts for the fact that we can attach a sequence of submaps (each two separated by an edge). Squaring the reciprocal accounts for the fact that in a quadrangulation Q we have |epQq| “ 2p|vpQq| ´ 2q. In equation (3.3.5), the term

2zp1`M pzqq 1´2zp1`M pzqq

block is a single edge. The multiplication

takes into consideration when the root

1 1´2zp1`M pzqq

in the first term accounts for

the extra submap attachment due to split of the root edge (recall the construction preceding Proposition 3.1.2).

Lemma 3.3.6. We have the following substitution relation between Cpzq and Bpzq: (3.3.7)

Cpzq “ BpU pzqq ¨ p1 ` Cpzqq .

To see that this identity gives a composition schema, note that it may equivˆ pzqq with B ˆ “ B{p1 ´ Bq. The multiplication alently be written as Cpzq “ BpU p1 ` Cpzqq accounts for the extra submap attachment due to the split of the root edge (see the construction preceding Proposition 3.1.3).

3.4. Airy Distribution for Quadrangulations In this section, we describe composition schemata for both rooted connected and 2-connected quadrangulations, and then establish corresponding distributional results.

57

3.4. AIRY DISTRIBUTION FOR QUADRANGULATIONS

58

Proposition 3.4.1. The triple pM, C, Hq is a map schema with (3.4.1)

α0 “

40 52{3 ¨ 15 9 7 , α“ , β“ , γ “ 1{3 . 15 27 28 5 ¨4

Proposition 3.4.2. The triple pC, B, U q is a map schema with (3.4.2)

5 211{2 ¨ 9 72{3 5 α0 “ , α “ 1{2 , β “ 1{3 , γ “ 1{3 . 7 5 ¨ 50 6 ¨2 42 ¨ 2

These two map schemata lead to the Airy distributional results stated in Propositions 3.1.4 and 3.4.3. Proof of Proposition 3.1.4. This now follows immediately from Corollary 3.3.3 and Proposition 3.4.2.



We will also need the following analogue of Proposition 3.1.4 for the largest 2-connected block of a general quadrangulation. Proposition 3.4.3. Let Qq Pu Qq , then for any A ą 0, uniformly over x P r´A, As, ` X \˘ βA pβxq p1 ` op1qq , P bpQq q “ 7q{15 ` xq 2{3 “ q 2{3 where β is given in Proposition 3.4.1. Proof. This follows from Corollary 3.3.3 and Proposition 3.4.1.



Before proving Propositions 3.4.1 and 3.4.2, we first establish a system of parameterization for M pzq, which is the key to showing that M pzq is singular with exponent 3{2 and to extracting the coefficients of M pzq, Cpzq and Hpzq. Lemma 3.4.4. Let ψM ptq “

tp2´9tq , p1´3tq2

let φM ptq “

1 , 1´3t

and let LM pzq be defined

by the implicit relation LM pzq “ zφM pLM pzqq, then M pzq “ ψM pLM pzqq . This parameterization is the one used in Goulden & Jackson [38]. It differs slightly from the original parameterization given by Tutte [65], but the two are related by a binational transformation.

3.4. AIRY DISTRIBUTION FOR QUADRANGULATIONS

59

Algebraic functions with such parameterization are called Lagrangean.

The

proof is a textbook application of the so-called quadratic method; presented in Section 3.9.2. A Lagrangean parametrization for M is derived in [16, Proposition 1]. The parameterization in Lemma 3.4.4 is slightly different, but the derivation is quite the same, and refer readers to that work the idea of the proof. (Also, in [16] the parameterization is stated for the generating function of general maps but this is equivalent, using Tutte’s angular bijection, to quadrangulations. See also Goulden & Jackson [38, Section 2.9] for a detailed explanation of the quadratic method for map enumeration.) One may inspect the Taylor expansion of ψM pLM pzqq at z “ 0 to conclude that this parametrization gives M pzq “ 2z ` 9z 2 ` 54z 3 ` 378z 4 ` O pz 5 q. Corollary 3.4.5 (Tutte [65]). (3.4.3)

(3.4.4)

LM pzq “

M pzq “

1 1 ´ p1 ´ 12zq1{2 , 6 6

` ˘ 1 4 8 ´ p1 ´ 12zq ` p1 ´ 12zq3{2 ` O p1 ´ 12zq2 . 3 3 3

In particular, M pzq is singular with exponent 3{2. Proof. Using Lemma 3.4.4, Lagrange inversion yields the explicit formulas LM pzq “

1 1 ´ p1 ´ 12zq1{2 , 6 6

M pzq “ ´1 `

1 p´p1 ´ 18zq ` p1 ´ 12zq3{2 q . 54z 2

Writing y “ 1´12z, the asymptotic expansion for M follows easily by rearrangement.  Implicit functional equations can be used to derive asymptotic expansions in great generality, even when no closed form is available, and we exploit this machinery in the current paper. We now sketch how the method is applied in our setting in slightly more detail, referring the reader to [16] and [36, VII.8] for a full exposition. Suppose we are given y defined by an implicit formula ypzq “ zφpypzqq, where φ is analytic, non-zero at 0, has non-negative Taylor coefficients, and has lim

xÑrφ

xφ1 pxq φpxq

ą 1,

where rφ P p0, 8s is the radius of convergence of φ. (In our case, φ will always in fact

3.4. AIRY DISTRIBUTION FOR QUADRANGULATIONS

be a rational function satisfying the preceding conditions.) Then, using Lagrange inversion, one obtains an asymptotic expansion of y around its dominant singularity (see [36, Section VI.7]). Given another function m expressible as mpzq “ ψpypzqq where ψ is a rational function whose radius of convergence is at least as large as that of y, this yields an asymptotic expansion for m as follows. First, we locate the radius of convergence for y. By [36, Theorem VI.6], we can expand ypzq as (3.4.5)

ypzq “ τ ´ l1{2 p1 ´ z{ry q

1{2

´ ¯ 3{2 ` l1 p1 ´ z{ry q ` O p1 ´ z{ry q ,

where the coefficients li{2 are to be determined for i P N, and ry and τ are determined by the equations τ φ1 pτ q ´ φpτ q “ 0, ry “ To determine l1{2 and l1 , let hptq “ ry ´

t . φptq

τ . φpτ q

Then hpτ q “ 0 “ h1 pτ q. Recall that

ypzq “ zφpypzqq. So expanding hpypzqq around τ yields 1 ´ z{ry “

hpypzqq ry

˘ ` h2 pτ q h3 pτ q pypzq ´ τ q2 ` pypzq ´ τ q3 ` O pypzq ´ τ q4 2ry 6ry ´ ¯ı2 h2 pτ q ” “ ´l1{2 p1 ´ z{ry q1{2 ` l1 p1 ´ z{ry q ` O p1 ´ z{ry q3{2 2ry ´ ¯ı3 ` ˘ h3 pτ q ” ` ´l1{2 p1 ´ z{ry q1{2 ` l1 p1 ´ z{ry q ` O p1 ´ z{ry q3{2 ` O pypzq ´ τ q4 6ry ˆ 2 ˙ ` ˘ h2 pτ q 2 h pτ q h3 pτ q 3 l1{2 p1 ´ z{ry q ´ 2 l1{2 l1 ` l1{2 p1 ´ z{ry q3{2 ` O p1 ´ z{ry q2 . “ 2ry 2ry 6ry



By comparing the coefficients of the terms p1 ´ z{ry q we obtain ˆ (3.4.6)

l1{2 “

2ry h2 pτ q

˙1{2

ˆ “

2φpτ q φ2 pτ q

˙1{2 ,

60

3.4. AIRY DISTRIBUTION FOR QUADRANGULATIONS

and by comparing the coefficients of the terms p1 ´ z{ry q3{2 we have (3.4.7)

l1 “ ´

3 h3 pτ ql1{2

6h2 pτ ql1{2

.

Now we use the expansion (3.4.5) to derive an expansion for mpzq around its dominant singularity rm . First, the equation mpzq “ ψpypzqq and the assumption that rψ ě ry together imply that rm “ ry . In the current work, we always have that ψ 1 pτ q “ 0 (indeed, this seems to generally be the case in compositional schemata involving maps); together with (3.4.5), a Taylor expansion of ψ around τ then yields mpzq “ ψpypzqq ´ ` ˘¯ “ ψ τ ´ l1{2 p1 ´ z{ry q1{2 ` l1 p1 ´ z{ry q ` O p1 ´ z{ry q3{2 ” ` ˘ı2 1 “ ψ pτ q ` ψ 2 pτ q ´l1{2 p1 ´ z{ry q1{2 ` l1 p1 ´ z{ry q ` O p1 ´ z{ry q3{2 2 ” ` ˘ ı3 1 3 ` ψ pτ q ´l1{2 p1 ´ z{ry q1{2 ` l1 p1 ´ z{ry q ` O p1 ´ z{ry q3{2 6 ` ˘ ` O p1 ´ z{ry q2 ˆ ˙ 1 2 1 3 2 3 2 “ ψ pτ q ` ψ pτ q l1{2 p1 ´ z{ry q ´ ψ pτ q l1{2 l1 ` ψ pτ q l1{2 p1 ´ z{ry q3{2 2 6 ` ˘ ` O p1 ´ z{ry q2 . We remark that the vanishing term ψ 1 pτ q “ 0 accounts for the shift of the singular exponent to 3{2. Using the compositional relation given in Lemma 3.3.5 together with the expansion of LM pzq given in Corollary 3.4.5, we show that Hpzq is also Lagrangean and expand Hpzq at its radius of convergence. We similarly deduce such kind of expansions for Cpzq, U pzq, and Bpzq.

Lemma 3.4.6. Table 3.1 gives Lagrangean parameterizations for M pzq, Hpzq, Cpzq, U pzq, Bpzq. Consequently, Hpzq, Cpzq, U pzq, and Bpzq each has radius of convergence and asymptotic expansion around rH , rC , rU , and rB as given in Table 3.2.

61

3.4. AIRY DISTRIBUTION FOR QUADRANGULATIONS

62

Proof. Let Hpzq be defined as in (3.3.3), and let ψM ptq be given in Lemma 3.4.4. Write t “ LM pzq, then ˙2

ˆ

1 1 ´ 2zp1 ` M pzqq

ˆ “z

1 1 ´ 2z p1 ` ψM pLM pzqqq

“ ´

tp´1 ` 3tq3 . p1 ´ 5t ` 8t2 q2

Hpzq “ z

˙2

Then taking ψH ptq as given by Table 3.1 yields Hpzq “ ψH pLM pzqq. The remaining parameterizations of Table 3.1 are established similarly, using (3.3.5) for Cpzq, and (3.3.7) for U pzq and Bpzq. The radius of convergence and expansions around the radius are derived using Lagrange inversion as in Corollary 3.4.5. Detailed derivation for the parameterizations of Tables 3.1 and 3.2 appear in Section 3.10.



f

φf

ψf

H

1 1´3t

tp´1`3tq ´ p1´5t`8t 2 q2

3

2 2

t2 p´1`5tq p´1`3tq3

2 2

´ tp´1`4tq p´1`3tq3

q C ´ p1´5t`8t p´1`3tq3 q U ´ p1´5t`8t p´1`3tq3

B

3

´ p´1`3tq p´1`4tq2

2

t2 p´1`5tq p´1`4tqp1´5t`8t2 q

Table 3.1. (Always with Lf pzq “ zφf pLf pzqq, where f is one of the functions H, C, U , and B.)

Remark. One of the fundamental facts of singularity analysis is that the radius of convergence of a generating function determines the exponential growth rate of the associated combinatorial family. Under Tutte’s angular bijection (see [65]), 2connected and simple quadrangulations respectively correspond to 2-edge-connected and 2-connected maps. In view of this, the values rC “ 27{196 and rB “ 4{27 agree with the known exponential growth rates for loopless, bridgeless maps Walsh & Lehman [66, (7)] and for 2-connected maps [16, Table 2] (noting that the coefficients

3.5. SIZES AND DIAMETERS OF PENDANT SUBMAPS

f

rf

H

1{12

expansion at rf 54 p1 ´ 12zq ` 343 p1 ´ 12zq3{2 ` O pp1 ´ 12zq2 q b 28 392 7 p1 ´ 196z{27q3{2 ` O pp1 ´ 196z{27q2 q ´ 135 p1 ´ 196z{27q ` 675 15 b 28 112 7 p1 ´ 196z{27q3{2 ` O pp1 ´ 196z{27q2 q ´ 135 p1 ´ 196z{27q ` 675 15 27 196

C 27{196

1 27

U 27{196

4 27

B

63

1 28

4{27

´

´

27 196

405 1372

p1 ´ 27z{4q `

? 9 3 98

p1 ´ 27z{4q3{2 ` O pp1 ´ 27z{4q2 q

Table 3.2

of the expansion for Bpzq are slightly different than in [16], because in that work a single loop is counted as a 2-connected map). Proofs of Propositions 3.4.1 and 3.4.2. We have verified that M pzq and Hpzq are singular with exponent 3{2 in Lemmas 3.4.5 and 3.4.6 respectively. The facts that HprH q “ rC and that U prU q “ rB are immediate from the values and expansions given in Table 3.2. Thus, pM, C, Hq and pC, B, U q are map schemata. The values claimed in (3.4.1) and (3.4.2) are then derived by routine arithmetic.



3.5. Sizes and Diameters of Pendant Submaps In this section, we first obtain a size bound for the decorations of the largest simple block in a uniform rooted 2-connected quadrangulation. Using this we then derive a corresponding diameter bound which leads to a “GH convergence version” of Proposition 3.1.1, shown in Section 3.5.1. Analogous tail bounds for uniform rooted quadrangulations are stated in Section 3.5.2. Proof of Proposition 3.1.5. Let λ : N Ñ N be a function tending to infinity with λprq ď

r1{3 . plog rq2

For k ď r ´ r2{3 λprq, the bound follows straightforwardly from

Proposition 3.1.4, Corollary 3.3.4, and the fact that the largest block is larger than or equal in size to the root block. We hereafter assume that r ´ r2{3 λprq ă k ď r. Note that for r large enough, r ´ r2{3 λprq ą r{2, so there must be less than one largest simple block of size k.

3.5. SIZES AND DIAMETERS OF PENDANT SUBMAPS

Let Rr Pu Rr , and note that P psbpRr q “ kq “

|Rr,k | . |Rr |

64

By Proposition 3.3.1 and

Lemma 3.4.6, |Rr | “ rz r´2 sCpzq “ Θp1q ¨ r´5{2 rC´r as r Ñ 8 with rC “

27 . 196

Each element of Rr,k may be constructed by first choosing S P Sk and a collection ř pMe , e P epSqq of rooted 2-connected quadrangulations and with ePepSq |epMe q| “ 2pr ´ kq; then attaching each Me to e P epSq to obtain a map R with r edges; and, finally, fixing a root edge e in R from among the p4r ´ 8q possible choices. This builds a map pR, eq P Rr,k , and any element of Rr,k may be so built. It follows that ¨

˛ ÿ

2k´4 ź

|Rr,k | ď |Sk | ¨ ˝

|Rxi |‚¨ p4r ´ 8q ,

px1 ,...,x2k´4 q i“1

where the sum is over non-negative integer vectors px1 , . . . , x2k´4 q with

ř iď2k´4

xi “

r ´ k. It is easily verified that for all s, t, |Rs ||Rt | ď |Rs`t´2 | ď |Rs`t |, so in the ś above sum we always have 2k´4 i“1 |Rxi | ď |Rr´k |. The number of summands is clearly less than p2k ´ 4qr´k , so we obtain |Rr,k | ď |Sk | ¨ p2k ´ 4qr´k ¨ |Rr´k | ¨ p4r ´ 8q . Recalling that |Sk | “ rz k´2 sBpzq, |Rr´k | “ rz r´k´2 sCpzq, this yields ´k |Rr,k | ď Θprq ¨ p2k ´ 4qr´k ¨ k ´5{2 ¨ rB ¨ pr ´ kq´5{2 ¨ rC´r`k ,

where rB and rC appear in Table 3.2. Altogether, for r ´ r2{3 λprq ă k ď r, |Rr,k | P psbpRr q “ kq “ ď Θprq ¨ p2k ´ 4qr´k ¨ k ´5{2 ¨ pr ´ kq´5{2 ¨ r5{2 ¨ |Rr | For large enough r we have r ´ k ă λprqr2{3 ď ˆ r´k

p2k´4q

“ exp ppr ´ kq ¨ logp2k ´ 4qq ď exp

r , plog rq2

ˆ

rC rB

˙k .

so for such r,

r ¨ logp2k ´ 4q plog rq2

˙

ˆ ď exp

r log r

˙ .

3.5. SIZES AND DIAMETERS OF PENDANT SUBMAPS

We have

rC rB

ă 1, so there exists b ą 0 such that

rC rB

65

ď exp p´bq. It follows that for

some positive constants c1 and c2 , ˆ

P psbpRr q “ kq ď p4r ´ 8q ¨ k

´5{2

´5{2

¨ pr ´ kq

¨r

5{2

r ¨ exp ´b ¨ k ` log r

˙

` ˘ ď c1 exp ´c2 rpk{r ´ 5{7q3 .



Proof of Proposition 3.1.6. For all positive integers r and x with x ď r ´ sprq ` 2 write Lr,x “ R P Rr,sprq : LpRq “ x

(

.

Fix ε P p0, 1{3q for the remainder of the proof. Letting Rr Pu Rr,sprq , `

(3.5.1)

P LpRr q ě r

2{3`ε

˘

“ |Rr,sprq |

´1

r´sprq`2 ÿ

|Lr,x | .

x“rr2{3`ε s

` ˘ Since sprq “ 5r{7 ` O r2{3 as r Ñ 8, by Proposition 3.1.4, (3.5.2)

` ˘ |Rr,sprq | “ Θ r´2{3 ¨ |Rr | “ Θpr´2{3 q ¨ r´5{2 ¨ rC´r “ Θpr´19{6 q ¨ rC´r .

Thus, it remains to bound |Lr,x |. Each element of Lr,x can be obtained by attaching some Rx P Rx to the largest simple block of some R P Rr´x`2,sprq , then possibly re-assigning the root edge. We therefore have (3.5.3)

|Lr,x | ď Θpr ¨ sprqq ¨ |Rr´x`2,sprq | ¨ |Rx |

as r Ñ 8; Θpsprqq accounts for choosing an edge of the largest simple block of R to attach Rx , while Θprq accounts for re-assinging the root edge. Then to bound |Lr,x |, it suffices to bound |Rr´x`2,sprq | and |Rx |. For large enough r and for all x P rr2{3`ε , r ´ sprq ` 2s, we have 5pr ´ x ` 2q{7 ` pr ´ x ` 2q2{3 ď sprq ď r ´ x ` 2. For x in this range, we may apply Proposition 3.1.5: we obtain that for some C 1 ą 0, ˜ ˆ ˙3 ¸ |Rr´x`2,sprq | sprq “ Op1q ¨ exp ´C 1 pr ´ xq ´ 5{7 . |Rr´x`2 | r´x

3.5. SIZES AND DIAMETERS OF PENDANT SUBMAPS

66

For all possible x, by Proposition 3.3.2 and Lemma 3.4.6 we have |Rr´x`2 | “ Θp1q ¨ pr ´ xq´5{2 rC´r`x , so ˜

ˆ

|Rr´x`2,sprq | “ Op1q ¨ pr ´ xq´5{2 ¨ rC´r`x ¨ exp ´C 1 pr ´ xq

˙3 ¸ sprq ´ 5{7 . r´x

Then (3.5.3) gives ˜

ˆ

|Lr,x | “ Opr ¨ sprqq ¨ x´5{2 ¨ pr ´ xq´5{2 ¨ rC´r ¨ exp ´C 1 pr ´ xq Since

sprq r

˙3 ¸ sprq ´ 5{7 . r´x

ě 5{7 ´ Cr´1{3 , we have for large r,

˜

ˆ

exp ´C 1 pr ´ xq

˙3 ¸ ´ ` ˘¯ sprq 1 ´2 2{3 3 ´ 5{7 ď exp ´C pr ´ xq 5x{7 ´ Cr . r´x

For r2{3`ε ď x ď r ´ sprq, and for large enough r, we thus have |Lr,x | ´ ` ˘¯ 1 ´2 2{3 3 “ Opr ¨ sprqq ¨ x ¨ pr ´ xq ¨ ¨ exp ´C pr ´ xq 5x{7 ´ Cr ´ ` ˘´2 ` 2{3`ε ˘3 ¯ “ Opr ¨ sprqq ¨ x´5{2 ¨ pr ´ xq´5{2 ¨ rC´r ¨ exp ´C 1 r ´ r2{3`ε 5r {7 ´ Cr2{3 ´5{2

´5{2

rC´r

(3.5.4) ˘ ` “ rC´r ¨ exp ´C 2 ¨ r3ε , for some C 2 ą 0. ` ˘ Finally, combining (3.5.1), (3.5.2), (3.5.4) and the fact that sprq “ 5r{7`O r2{3 , there exist positive constants c1 , c2 and c3 “ c3 pεq such that `

P LpRr q ě r

2{3`ε

˘

“ |Rr,sprq |

´1

r´sprq`2 ÿ r2{3`ε

x“r

|Lr,x | ď c1 exp p´c2 rc3 q .



s

3.5.1. Diameters of submaps pendant to the largest simple block. We want to apply [28] to obtain a diameter bound, but first we need to transfer the diameter tail bound from [28] to the setting of 2-connected quadrangulations.

3.5. SIZES AND DIAMETERS OF PENDANT SUBMAPS

67

Lemma 3.5.1. Let Rr Pu Rr , then there exist positive constants x0 , c1 and c2 such that for all x ą x0 , ` ˘ P diam pRr q ą xr1{4 ď c1 r2{3 exp p´c2 xq . Proof. For q P N, let Qq Pu Qq . Given that bpQq q “ r, RpQq q has the same distribution as Rr . So for all q ě r and x ą 0, we have ˇ ` ˘ ` ˘ P diam pRr q ą xr1{4 “ P diam pRpQq qq ą xr1{4 ˇ bpQq q “ r ˇ ` ˘ ď P diam pQq q ą xr1{4 ˇ bpQq q “ r ` ˘ P diam pQq q ą xr1{4 . ď P pbpQq q “ rq Now let q “ t15r{7u, then xr1{4 ě xp7{15q1{4 q 1{4 , so by Proposition 3.1.7, there exist positive constants x0 , C1 , C2 such that for all x ą x0 , ` ˘ P diam pQq q ą xr1{4 ď C1 expp´C2 xq . On the other hand, by Proposition 3.4.3, there exists C3 ą 0 such that for all r P N, P pbpQq q “ rq ě C3 r´2{3 . Altogether, we have ` ˘ C1 exp p´C2 xq P diam pRr q ą xr1{4 ď C3 r´2{3 Then setting c1 “ C1 {C3 and c2 “ C2 concludes the proof.



Proof of Proposition 3.1.8. Fix a positive integer r and let k P N with k ď mintsprq, r ´ sprqu. Let Rr “ pRr , er q Pu Rr,sprq , write Sr “ SpRr q, let e1 be the ăRr -minimal oriented edge of Sr , and write R1r “ pRr , e1 q. It follows from Proposition 3.1.2 that R1r uniquely decomposes as pSr , Θq P S ˆ R|epSr q|`1 . Write Θ “ pΘ0 , Θ1 , . . . , Θ|epSpRr qq| q; recall that Θ has two entries corresponding to the root edge.

3.5. SIZES AND DIAMETERS OF PENDANT SUBMAPS

68

For any 0 ď i ď |epSr q|, given that |vpΘi q| “ k, Θi is uniformly distributed over Rk . By Lemma 3.5.1 and since k ď r, there exist positive constants x0 , c1 and c2 such that for all x ě x0 , and for all 0 ď i ď |epSr q|, ˇ ` ˘ P diampΘi q ě xk 1{4 ˇ |vpΘi q| “ k ď c1 r2{3 exp p´c2 xq .

(3.5.5)

Note that |epSr q| “ 2sprq ´ 4 and recall that DpRr q “ maxpdiampΘi q : 0 ď i ď 2sprq ´ 4q. Fix ε P p0, 1{7q. Using a union bound, ` ˘ P DpRr q ě r5{24 » fi 2sprq´4 tr2{3`ε ÿ ÿ u ` ˘ ` ˘ – P diam pΘi q ě r5{24 , |vpΘi q| “ k ` P |vpΘi q| ě r2{3`ε fl ď i“0

k“1

2sprq´4 ÿ

ď

»

tr2{3`ε ÿ u

– i“0

fi ˇ ´ ¯ ` ˘ ˇ P diam pΘi q ě r5{24 ˇ |vpΘi q| “ k ` P |vpΘi q| ě r2{3`ε fl .

k“1

By (4.7.1), for k ď r2{3`ε and for each 0 ď i ď 2sprq ´ 4, ˇ ´ ¯ ˘ ` 5{24 ˇ P diam pΘi q ě r ˇ |vpΘi q| “ k ď c1 r2{3 exp ´c2 r5{24 k ´1{4 ˘ ` ď c1 r2{3 exp ´c2 r1{24´ε{4 . Finally, by Proposition 3.1.6, there exist positive constants k1 , k2 and k3 “ k3 pεq such that for each 0 ď i ď 2sprq ´ 4, ` ˘ ` ˘ ` ˘ P |vpΘi q| ě r2{3`ε ď P LpRr q ě r2{3`ε ď k1 exp ´k2 rk3 ; combining the preceding 3 inequalities and using that sprq ď r and that 1{24´ε{4 ą 1{168 yields ` ˘ “ ` ˘ ` ˘‰ P DpRr q ě r5{24 ď p2sprq ´ 3q r2{3`ε ¨ c1 r2{3 exp ´c2 r1{24´ε{4 ` k1 exp ´k2 rk3 ` ` ˘˘ ` ` ˘˘ “ O r7{3`ε exp ´c2 r1{168 ` O r ¨ exp ´k2 rk3 . By choosing the constants C1 , C2 and C3 carefully, we may conclude the proof.



3.5. SIZES AND DIAMETERS OF PENDANT SUBMAPS

69

Given the diameter bound, we immediately have the “GH convergence version” of Proposition 3.1.1:

Proposition 3.5.2. Let Rr Pu Rr,sprq and write Sr “ SpRr q, then as r Ñ 8, (3.5.6) ˜˜ vpRr q,

ˆ

21 40r

¸

˙1{4 ¨ dRr

˜ ,

vpSr q,

ˆ

21 40r

¸¸

˙1{4 ¨ dSr

d

Ñ ppM, D1˚ q, pM, D1˚ qq

in distribution for the Gromov-Hausdorff topology.

Proof. For any compact metric space X “ pX, dq and any subspace Y “ pY, d|Y ˆY q we have dGH pX, Yq ď supxPX dpx, Y q. By Proposition 3.1.8, p

sup r´1{4 ¨ dRr pv, Sr q Ñ 0 , vPvpRr q

and the result follows.



3.5.2. Analogous results for the largest 2-connected block. By analogy to Propositions 3.1.6 and 3.1.8, we have the following bounds for the submaps pendant to the largest 2-connected block in a uniform quadrangulation. Given Qq “ pQq , eq P Qq , write Rq “ RpQq q, let e1 be the ăQq -minimal oriented edge of Rq , and write Q1q “ pQq , e1 q. By Proposition 3.1.3, Q1q uniquely decomposes as pRq , ppLi , bi q : 0 ď i ď 2|epRq q| ´ 4qq , where Li “ pMi,j : 1 ď j ď li q P Qli and bi “ pbi,j : 1 ď j ď li q P t0, 1uli , and pli : 0 ď i ď 2|epRq q| ´ 4q are suitable non-negative integers. Write (3.5.7)

L1 pQq q “ max p|vpMi,j q| : 0 ď i ď 2|epRq q| ´ 4, 1 ď j ď li q ,

(3.5.8)

D1 pQq q “ max pdiampMi,j q : 0 ď i ď 2|epRq q| ´ 4, 1 ď j ď li q .

3.6. EXCHANGEABLE DECORATIONS

Proposition 3.5.3. For all ε P p0, 1{3q, there exist positive constants c1 , c2 , and c3 “ c3 pεq such that, if Qq Pu Qq,rpqq , ` ˘ P L1 pQq q ě q 2{3`ε ď c1 exp p´c2 q c3 q . Proposition 3.5.4. Let Qq Pu Qq,rpqq , then there exist positive constants C1 , C2 , and C3 such that ˘ ` ˘ ` P D1 pQq q ě q 5{24 ď C1 exp ´C2 q C3 . We omit proofs for the above propositions since the arguments are very similar as those for Propositions 3.1.6 and 3.1.8.

3.6. Exchangeable Decorations This section provides bounds on the Prokhorov distance between three sorts of measures on the vertices of a graph: the uniform measure, the degree-biased measure, and measures obtained by assigning vertices exchangeable random masses. In subsequent sections, these bounds help control the GHP distance between a map and its largest block. Recall from the introduction that for a map G, and c ą 0, cG denotes the measured metric space pvpGq, c ¨ dG , µG q. Given a map G, write degG pvq for the degree of v P vpGq in G; the degree-biased measure on G is the measure µB G on vpGq ř satisfying µB G pSq “ vPS degG pvq{p2|epGq|q.

Lemma 3.6.1. For any quadrangulation Q and any ε ą 0, with µG and µB G viewed as measures on εQ, we have dP pµQ , µB Q q ď maxpε, 1{|vpQq|q. Proof. Let n be the number of faces of Q, so that |vpQq| “ n ` 2 and |epQq| “ 2n. Fix V Ĺ vpQq (the remaining case is trivial). Write V ` “ tv P vpQq : dQ pv, V q ď

70

3.6. EXCHANGEABLE DECORATIONS

71

1u. We claim that ` µB Q pV q ě µQ pV q ´

(3.6.1)

1 . |vpQq|

ε If this is so then in εQ we obtain µQ pV q ď µB Q pV q ` 1{|vpQq|; since V was arbitrary,

the lemma then follows easily. We now prove (3.6.1). First suppose QrV s is connected, and write p “ |V |. If p “ 1 then the inequality is easily checked. If p ě 2 then view QrV s as a quadrangulation with boundaries; let ř the boundaries have lengths `1 , . . . , `k and write ki“1 `i “ `. We have k ě 1 since V ‰ vpQq. A face f of Q is an internal face of QrV s if all vertices of f lie in V ; it is a boundary face of QrV s if some edge of QrV s is incident to f , but not all edges incident to f belong to QrV s. Writing i for the number of internal faces of QrV s, Euler’s formula straightforwardly yields p “ i ` 2 ` `{2 ´ k. Furthermore, if f is a boundary face of QrV s then all edges of f lie within QrV ` s. Now, a boundary face can be incident to at most two edges of QrV s, so QrV s must have at least `{2 boundary faces. It follows that ÿ vPV `

Since

ÿ

degQ pvq ě f

ř vPvpQq

internal to

4 ě 4pi ` `{2q “ 4pp ` k ´ 2q ě 4pp ´ 1q. QrV

`s

degQ pvq “ 2|epQq| “ 4n, it follows that

µB Q pV q ě

p´1 n ` 2 ´ 2p 1 “ µQ pV q ´ ě µQ pV q ´ . n npn ` 2q |vpQq|

Finally, if QrV s is not connected, the same argument applied component-wise yields the same bound.



3 1{4 Since both 1{s Ñ 0 and p 8s q Ñ 0 as s Ñ 8, the following is immediate.

Corollary 3.6.2. For Ss P Ss , with µSs and µB Ss viewed as measures on

` 3 ˘1{4 8s

Ss ,

we have dP pµSs , µB Ss q Ñ 0 as s Ñ 8. ř In what follows, for a vector x “ px1 , . . . , xk q P Rk write |x|p “ p ki“1 xpi q1{p . Suppose that G “ pG, eq is a rooted map. Enumerate the edges of G as e1 , . . . , em ,

3.6. EXCHANGEABLE DECORATIONS

72

where m “ |epGq|, and let e0 be a second copy of the root edge e. (This makes sense even if G is random, as long as it is possible to specify a canonical way to order the edges of G; for example, we may use the order ăG described in the introduction.) For each 0 ď i ď m, let wi be a uniformly random endpoint of ei . Let n “ pn0 , . . . , nm q be a vector of non-negative real numbers with |n|1 ą 0. Define a (random) probability measure νGn on vpGq as follows: for V Ă vpGq, let νGn pV q “

(3.6.2)

ÿ 1 ¨ ni . |n|1 ti:w PV u i

If one views pwi : 0 ď i ď 2s ´ 4q as attachment locations for pendant submaps, and n as listing the masses of these submaps, then νGn is the probability measure assigning each vertex v a mass proportional to the total mass of submaps pendant to v. Lemma 3.6.3. Let G “ pG, eq have |epGq| “ m and let n “ pn0 , . . . , nm q be an exchangeable random vector of non-negative real numbers with |n|2 strictly positive. Then for any V Ă vpGq, ˇ ˙ ˆ ˙ ˇ 1 2t2 2t n B ˇ ` |n|2 ď 4 exp ´ 2 . P |νG pV q ´ µG pV q| ą |n|1 m ` 1 ˇ |n|2 ˆ

In the proof, we will use the following result of Aldous [7, Proposition 20.6], which informally says that partial sums constructed by sampling without replacement may be obtained by first sampling with replacement and then taking a suitable projection. Proposition 3.6.4 ([7], Proposition 20.6). Fix x1 , . . . , xm P R and k P t1, . . . , mu, let σ be a uniformly random permutation of t1, . . . , mu, and let I1 , . . . , Ik be independent and uniform on t1, . . . , mu. Then there exists a pair of random variables pX, Y q such that E rY |Xs “ X and d

X“

k ÿ j“1

xσpjq ,

d

Y “

k ÿ

xI j .

j“1

Aldous [7] notes the following consequence of the preceding proposition, which is what we will in fact use.

3.6. EXCHANGEABLE DECORATIONS

73

Corollary 3.6.5 ([64], Theorem 2). Under the conditions of Proposition 3.6.4, for all continuous convex functions φ : R Ñ R, E rφpXqs ď E rφpY qs . Proof of Lemma 3.6.3. Given V Ă vpGq, write Be V for the edge boundary of S, i.e., the set of edges e1 P epGq with one endpoint in V and one in V c . By definition, for 0 ď j ď m, the vertex wj is a uniformly random endpoint of ej . We have ř

tj:ej PGrV su nj `

νGn pV q “

(3.6.3)

ř tj:ej PBe V u

1rwj PV s nj

|n|1

.

We now show that νGn pV q is concentrated using Proposition 3.6.4. Independently for each j ě 1 let Ij Pu t0, . . . , mu. By the exchangeability of n, it follows that for any continuous convex φ : R Ñ R, » ˆ ÿ E –φ

fi ˙ nj

»

fi

ˆ fl ď E –φ

tj:ej PGrV su

˙ ÿ

nIj fl .

tj:ej PGrV su

Also by exchangeability, fi

» ÿ

nIj fl “ |n|1 ¨

E– tj:ej PGrV su

Taking φpxq “ ecx for c –

|epGrV sq| . m`1

4t |n|22

and applying Markov’s inequality as in [54, Theorem ř 2.5] yields Hoeffding’s inequality-type bounds for tj:ej PGrV su nIj : ¨ ˇ ˇ ˝ P ˇ

nj |epGrV sq| ˇˇ t ´ ˇą |n|1 m`1 |n|1 PGrV su

ÿ tj:ej

¨ ˇ ˇ “ P ˝ˇ

˛ ˇ ˇ ˇ |n|2 ‚ ˇ

˛ ˇ ˇ ˇ |epGrV sq| ˇ nj ´ |n|1 ¨ ˇ ą t ˇˇ |n|2 ‚ m ` 1 tj:ej PGrV su » ´ ˇ ÿ |epGrV sq| ˇˇ¯ ˇ ď e´ct ¨ E –exp c ¨ ˇ nj ´ |n|1 ¨ ˇ m`1 tj:e PGrV su ÿ

j

fi ˇ ˇ ˇ |n|2 fl ˇ

3.7. PROJECTION OF MASSES IN RANDOM QUADRANGULATIONS

» ´ ˇ ˇ ´ct – ď e ¨ E exp c ¨ ˇ

tj:ej

ˆ ď 2 exp ´

2t2 |n|22

|epGrV sq| ˇˇ¯ nIj ´ |n|1 ¨ ˇ m`1 PGrV su

ÿ

74

fi ˇ ˇ ˇ |n|2 fl ˇ

˙ .

The random variables 1rwj PV s are iid Bernoullip1{2q, so a reprise of the argument yields ¨ ˇ ˇ P ˝ˇ

ÿ 1rwj PV s nj |Be V | ˇˇ t ´ ˇą |n|1 2pm ` 1q |n|1 tj:e PB V u j

e

˛ ˇ ˆ ˙ 2 ˇ 2t ˇ |n|2 ‚ ď 2 exp ´ . ˇ |n|2 2

We have µB G pV q “

ÿ 1 2|epGrV sq| ` |Be V | degpvq “ , 2|epGq| vPV 2m

so ˇ ˇ ˇ B ˇ |epGrV sq| |B V | e ˇµG pV q ´ ˇď 1 . ´ ˇ m`1 2pm ` 1q ˇ m ` 1 Considering (3.6.3), we then have ˆ P

|νGn pV

¨ ˇ ˇ ď P ˝ˇ



ÿ tj:ej PBe

¨ ˇ ˇ ` P ˝ˇ

µB G pV

ˇ ˙ 1 ˇˇ 2t ` |n|2 q| ą |n|1 m ` 1 ˇ

1rwj PV s nj t |Be V | ˇˇ ´ ˇą |n|1 2pm ` 1q |n|1 Vu

ÿ nj |epGrV sq| ˇˇ t ´ ˇą |n|1 m`1 |n|1 tj:e PGrV su

˛ ˇ ˇ ˇ |n|2 ‚ ˇ

˛ ˇ ˇ ˇ |n|2 ‚ . ˇ

j

Now combine the three probability inequalities.



It is easily seen that the above lemma applies even for random sets V , so long as V is independent of the randomness used to select the endpoints wi of the edges; we will use this in what follows.

3.7. Projection of Masses in Random Quadrangulations In this section we apply Lemma 3.6.3 to study projection of masses in large random quadrangulations, and in particular to prove Proposition 3.1.1. We begin

3.7. PROJECTION OF MASSES IN RANDOM QUADRANGULATIONS

75

by stating a straightforward corollary of Lemma 3.6.3. For a metric space X “ pX, dq and x P X write Bpx, r; Xq “ ty : dpx, yq ă ru. Corollary 3.7.1. For s P N let ns “ pns,0 , . . . , ns,2s´4 q be an exchangeable random vector of non-negative real numbers. Let Ss Pu Ss , and for v P vpSs q write Bpv, rq “ Bpv, r¨s1{4 ; Ss q. Conditional on Ss , let U and U 1 be independent, uniformly random elements of vpSs q. If |ns |1 Ñ 8 and |ns |2 {|ns |1 Ñ 0 then for all x ě 0, ˇ ˇ n ˇ ˇν s pBpU, xqq ´ µB Ss pBpU, xqq Ñ 0 , Ss

(3.7.1)

(3.7.2)

ˇ n ˇ 1 ˇν s pBpU, xq X BpU 1 , xqq ´ µB ˇ Ss pBpU, xq X BpU , xqq Ñ 0 Ss

in probability as s Ñ 8. Proof. Fix x ě 0. We assumed that |ns |1 Ñ 8 and |ns |2 {|ns |1 Ñ 0; we may therefore choose a sequence tpsq such that tpsq{|ns |1 Ñ 0 and tpsq{|ns |2 Ñ 8. Now take V “ BpU, xq. Recalling that |epSs q| ` 1 “ 2s ´ 3, for any ε ą 0, Lemma 3.6.3 gives ˇ ˙ ˇ ˇ ě ε ˇ |n|2 ´ lim sup P sÑ8 ˇ ˆ ˙ ˇ 2tpsq 1 ns B ˇ ď lim sup P |νSs pV q ´ µSs pV q| ě ` |n|2 |ns |1 2s ´ 3 ˇ sÑ8 ˆ ˙ 2tpsq2 ď lim sup 4 exp ´ |ns |22 sÑ8 ˆ

|νSnss pBpU, xqq

µB Ss pBpU, xqq|

“ 0, which is (3.7.1). To prove (3.7.2) take V “ BpU, xq X BpU 1 , xq and argue similarly.  Corollary 3.7.2. Under the assumptions of Corollary 3.7.1, with νSnss and µB Ss ` 3 ˘1{4 viewed as measures on 8s Ss , we have dP pµSs , νSnss q Ñ 0 in probability as s Ñ 8. p

ns Proof. By Corollary 3.6.2, it suffices to show that dP pµB Ss , νSs q Ñ 0.

To

achieve this, we use Corollary 3.7.1 and the compactness of the Brownian map

3.7. PROJECTION OF MASSES IN RANDOM QUADRANGULATIONS

m8 “ pM, D1˚ , µ1 q. For the remainder of the proof we abuse notation by writing ns µs “ µB Ss and νs “ νSs , for readability.

´ ¯ ` 3 ˘1{4 Fix ε ą 0. By [4, Theorem 1], the triple vpSs q, 8s dSs , µSs converges in distribution to m8 as r Ñ 8. Since m8 is almost surely compact and µ a.s. has support M, if pUi : i P Nq are independent with law µ then we almost surely have ˜ ¸ k ď K8 :“ inf k P N : BpUi , ε; m8 q “ M ă 8. i“1

For s P N, let pUs,i : i P Nq be independent with law µSs , and let ˜ Ks “ inf

kPN:

k ď

¸ BpUs,i , ε; p3{8sq1{4 Ss q “ vpSs q

.

i“1

The aforementioned distributional convergence and the a.s. finiteness of K8 together imply that there exists K P N such that for all s P N, P pKs ą Kq ă ε. For i ě 1 let ´ ¯ Bi “ B Us,i , ε; p3{p8sqq1{4 Ss .

(3.7.3)

Let A1 “ B1 , and for i ą 1 let Ai “ Bi z

Ťi´1 j“1

Bj . Then A1 , . . . , AKs is a covering of

vpSs q by disjoint sets. Suppose that dP pµs , νs q ą ε. Then there exists a set S Ă vpSs q such that either µs pS ε q ă νs pSq ´ ε or νs pS ε q ă µs pSq ´ ε. Since A1 , . . . , AKs partition vpSs q, there is j such that either µs pS ε X Aj q ă νs pS X Aj q ´ ε{Ks ď νs pAj q ´ ε{Ks or νs pS ε X Aj q ă µs pS X Aj q ´ ε{Ks ď µs pAj q ´ ε{Ks . For one of these to occur we must have S X Aj ‰ H. Since Aj has radius at most ε, it follows that Aj Ă S ε . Thus, either µs pAj q ă νs pAj q ´ ε{Ks

or

νs pAj q ă µs pAj q ´ ε{Ks .

76

3.7. PROJECTION OF MASSES IN RANDOM QUADRANGULATIONS

77

This yields the bound P pdP pµs , νs q ą εq ď P p|µs pAj q ´ νs pAj q| ą ε{Ks for some 1 ď j ď Ks q K ÿ

ď

P p|µs pAj q ´ νs pAj q| ą ε{Ks , Ks ď Kq ` P pKs ą Kq

j“1

(3.7.4)

K ÿ

ď

P p|µs pAj q ´ νs pAj q| ą ε{Kq ` ε .

j“1

By the triangle inequality, for all 1 ď i ď K, |µs pAi q ´ νs pAi q| ˇ ` ˘ ` ˘ˇ i´1 ˇ “ ˇµs Bi z Yi´1 j“1 Bj ´ νs Bi z Yj“1 Bj ˇ ` ˘ ` ˘ˇ i´1 i´1 ď |µs pBi q ´ νs pBi q| ` ˇµs Bi X Yj“1 Bj ´ νs Bi X Yj“1 Bj ˇ ď |µs pBi q ´ νs pBi q| `

i´1 ÿ

|µs pBi X Bj q ´ νs pBi X Bj q| ,

j“1

where the last sum equals 0 in the case i “ 1. Recalling the definitions of the Bi from (3.7.3), the preceding inequality and Corollary 3.7.1 imply that for each fixed i ě 1, |µs pAi q ´ νs pAi q| Ñ 0 in probability as s Ñ 8. Combining this with (3.7.4), we obtain lim sup P pdP pµs , νs q ą εq ď ε . sÑ8

Since ε ą 0 was arbitrary, this completes the proof.



We are almost ready to prove Proposition 3.1.1; before doing so we state two easy facts, which each provide bounds on the GHP distance between a measured metric space and an induced (in some sense) subspace. The first fact is immediate from the definition of dGHP .

3.7. PROJECTION OF MASSES IN RANDOM QUADRANGULATIONS

Fact 3.7.3. Fix a measured metric space X “ pX, d, µq and Y Ă X, and let µY be a non-negative finite Borel measure on pY, dY q, where dY “ d|Y ˆY is the induced metric. Write Y “ pY, dY , µY q. Then dGHP pX, Yq ď maxpdH pX, Yq, dP pµ, µY qq. The second fact informally says that in a compact measured metric space, projecting onto an ε-net does not change the space very much (in the GHP sense). The proof is left to the reader. Fact 3.7.4. Let X “ pX, d, µq be a compact measured metric space, and let S Ă X be finite so that there exists ε ą 0 with X Ă S ε . Let pXs : s P Sq be Ť measurable subsets of X such that sPS Xs “ X, that µpXs X Xs1 q “ 0 for s ‰ s1 , and that Xs Ă Bps, ε; Xq for all s P S. Define a measure ν on S by νpsq “ µpXs q for any s P S, and let S “ pS, d|SˆS , νq. Then dGHP pX, Sq ď ε . Proof of Proposition 3.1.1. For r P N, let Rr Pu Rr,sprq , and let Sr “ SpRr q. Write Rr “ pRr , er q, and let e1 be the ăRr -minimal oriented edge of Sr . Next, apply the bijection of Proposition 3.1.2 to the map R1r “ pRr , e1 q: this decomposes Rr into Sr together with a sequence pΘi : 0 ď i ď 2sprq ´ 4q of submaps of Rr . Let nr,0 “ |epΘ0 q|1r|epΘi q|ą1s , and for 1 ď i ď 2sprq ´ 4 let nr,i “ 1 ` |epΘi q|1r|epΘi q|ą1s . Then let nr “ pnr,0 , . . . , nr,2sprq´4 q and construct the measure νSnrr as in (3.6.2): for 0 ď i ď 2sprq ´ 4, wi is a random endpoint of ei , and νSnrr pvq “

ÿ 1 nr,i . 2r ´ 4 ti:w “vu i

(The difference of 1 in the definition of nr,0 accounts for the fact that when reconstructing Rr from Sr and the Θi , we identify two copies of the root edge; the fact that 2r ´ 4 is the correct normalization follows from (3.1.1).) We have |nr |1 “ 2r ´ 4 Ñ 8 as r Ñ 8. Furthermore, if LpRr q ď r3{4 then nr,i ď 2r3{4 ´ 3 for all i, so |nr |2 {|nr |1 “ Opr´1{8 q Ñ 0. By Proposition 3.1.6, we ` ˘ p have P LpRr q ď r3{4 Ñ 1, so |nr |2 {|nr |1 Ñ 0.

78

3.7. PROJECTION OF MASSES IN RANDOM QUADRANGULATIONS

Corollary 3.7.2 now implies that dP pµSr , νSnrr q Ñ 0 as r Ñ 8, with the measures ` 21 ˘1{4 viewed as living on 40r Sr . For Borel measures µ, ν on a metric space pX, dq, we have dGHP ppX, d, µq, pX, d, νqq “ dP pµ, νq, so (3.7.5)

˘˘ p ` 21 1{4 ` 21 1{4 q Sr , vpSr q, p 40r q ¨ dSr , νSnrr Ñ 0 . dGHP p 40r

21 1{4 We now bound the distance from p 40r q Rr to the latter space. It is convenient

to work with a graph with edge lengths rather than a finite measured metric space. More precisely, view each edge e of Rr as an isometric copy Ie of the unit interval r0, 1s, endowed with the rescaled Lebesgue measure p2r ´ 4q´1 ¨ LebIe , and write R1 “ pRr1 , d1r , µ1r q for the resulting measured metric space. We then have µ1r pRr1 q “ ř p2r ´ 4q´1 ¨ ePepRr q LebIe pIe q “ 1. We may naturally identify vpRr q with the set of endpoints of edges in R1 , and this is an isometric embedding in that with this identification we have dRr “ d1r |vpRr q . Furthermore, the degree-biased measure µB Rr may be obtained by projection onto 1 1 vpRr q: for v P vpRr q we have µB Rr pvq “ µr pBpv, 1{2; R qq “ degRr pvq{p2p2s ´ 4qq. By

Fact 3.7.4, it follows that for any ε ą 0, (3.7.6)

dGHP pεR1 , pvpRr q, εdRr , µB Rr qq ď ε.

The space Sr “ pvpSr q, dSr q is likewise isometrically embedded within R1 , and we may also obtain the measure νSnrr by projection. To do so, let Ei “ epΘi q for 1 ď i ď 2sprq ´ 4, let E0 “ EpΘ0 qzte1 u, and for v P vpSr q let Xv “

ď

ď

Ie .

ti:wi “vu ePEi

Then νSnrr pvq “ µ1r pXv q. Furthermore, pXv : v P vpSr qq covers Rr1 and µ1r pXu XXv q “ 0 for u ‰ v since edges only intersect at their endpoints. Recalling the definition of DpRr q from Section 3.1.2, for any v P V we have Xv Ă Bpv, DpRr q; R1 q. It follows from Fact 3.7.4 that for all ε ą 0, (3.7.7)

dGHP pεR1 , pvpSr q, εdSr , νSnrr qq ď ε ¨ DpRr q .

79

3.8. PROOFS OF THE MAIN THEOREMS

80

We always have DpRr q ě 1, so combining (3.7.6), (3.7.7) gives dGHP ppvpSr q, εdSr , νSnrr q, pvpRr q, εdRr , µB Rr qq ď 2ε ¨ DpRr q Using Lemma 3.6.1 to bound dGHP ppvpRr q, εdRr , µB Rr q, εRr q, the triangle inequality then gives (3.7.8)

dGHP ppvpSr q, εdSr , νSnrr q, εRr q ď 2ε ¨ DpRr q ` max pε, 1{rq . p

By Proposition 3.1.8, r´1{4 DpRr q Ñ 0, and (3.7.5) then implies that ` 21 1{4 ˘ p 21 1{4 dGHP p 40r q Sr , p 40r q Rr Ñ 0 . Since

3 8sprq

d

¨ Sr Ñ m8 as r Ñ 8, and

3 8sprq

21 “ p1 ` op1qq 40r , the result follows.



3.8. Proofs of the Main Theorems Recall that K is the set of measured isometry classes of compact metric spaces, and that GHP convergence refers to convergence in the the Polish space pK, dGHP q.

Proof of Theorem 3.1.1. Let g : K Ñ R be a bounded continuous function, ` 21 ˘1{4 and write }g} “ sup |g| ă 8. Recall that Rr Pu Rr , and let Mr “ 40r Rr . We show that (3.8.1)

E rgpMr qs Ñ E rgpm8 qs

as r Ñ 8; the result then follows by the Portmanteau theorem. The proof of (3.8.1) is simply summarized: average over the size of sbpRr q. The details are also fairly straightforward. Fix ε P p0, 1{2q with ε ă 1{}g}, let A be the Airy density and let β given by Proposition 3.1.4. Then fix Cε ą 0 large ş Cε enough that ´C βA pβxq dx ą 1 ´ ε. Recall from the introduction sprq satisfies ε |sprq ´ 5r{7| ď Cr2{3 for large r. The constant C was fixed but arbitrary, so we may assume that C ą Cε .

3.8. PROOFS OF THE MAIN THEOREMS

81

Next, for r, s P N with s ď r, let Rr,s Pu Rr,s and write Mr,s “

` 21 ˘1{4 40r

Rr,s . We

claim that (3.8.2)

sup

|gpMr,s q ´ gpm8 q| Ñ 0

tsPN:|s´5r{7|ďCε r2{3 u

as r Ñ 8. Indeed: otherwise we may find a sequence pˆ sprq, r ě 1q such that |ˆ sprq ´ 5r{7| ď Cε r2{3 ă Cr2{3 with lim suprÑ8 |gpMr,ˆsprq q ´ gpm8 q| ‰ 0. By the Portmanteau theorem, this implies that Mr,ˆsprq does not converge in distribution to m8 , contradicting Proposition 3.1.1. This establishes (3.8.2). Now for each r P N, let Er “ |sbpRr q ´ 5r{7| ď Cε r2{3

(

.

Recalling the definition of δs p¨q from Theorem 3.1.4, it follows from Proposition 3.1.4 and a Riemann approximation that for large enough r, ÿ

P pEr q “ p1 ` op1qq

tsPN:|s´5r{7|ďCε r2{3 u

βA pβδs pqqq r2{3

ż Cε βA pβsq ds

“ p1 ` op1qq ´Cε

ą 1 ´ 2ε. Then for large enough r, (3.8.3)

ˇ “ ‰ˇ ˇE rgpMr qs ´ E gpMr q 1rEr s ˇ ď P pErc q }g} ă 2ε}g} .

ˇ “ ˇ ‰ We now show that ˇE gpMr q 1rEr s ´ E rgpm8 qsˇ is also small. The conditional law of Rr given that sbpRr q “ s is identical to that of Rr,s , so “ ‰ E gpMr q 1rEr s “

ÿ

P psbpRr q “ sq E rgpMr,s qs .

tsPN:|s´5r{7|ďCε r2{3 u

By the triangle inequality, we therefore have ˇ “ ˇ ‰ ˇE gpMr q 1rEr s ´ E rgpm8 qsˇ

3.8. PROOFS OF THE MAIN THEOREMS

ÿ

P psbpRr q “ sq ¨ |E rgpMr,s qs ´ E rgpm8 qs|

ď tsPN:|s´5r{7|ďCε

(3.8.4)

r2{3 u

sup

ď

|gpMr,s q ´ gpm8 q| .

tsPN:|s´5r{7|ďCε r2{3 u

This tends to 0 by (3.8.2), which with (3.8.3) gives lim suprÑ8 |E rgpMr qs´E rgpm8 qs | ď 2ε}g}. Since ε ą 0 was arbitrary, this establishes (3.8.1) and completes the proof.  In the remaining proofs, we use the following simple fact. Recall the definition of L1 from (3.5.7). Fact 3.8.1. Let Qq P Qq , write Rq “ RpQq q and Sq “ SpQq q. Note that if Sq ‰ SpRq q, then sbpQq q ě sbpRq q, and it follows that Qq contains at least two 2-connected blocks of size at least sbpRq q, implying that L1 pQq q ě sbpRq q. Proof of Theorem 3.1.4. Fix q P N and write Rq “ RpQq q. Let R Pu Rrpqq . d

Given that bpQq q “ rpqq, Rq “ R, so P pbpQq q “ rpqq, sbpRq q “ sprpqqqq ˇ ` ˘ “ P sbpRq q “ sprpqqq ˇ bpQq q “ rpqq ¨ P pbpQq q “ rpqqq “ P psbpRq “ sprpqqqq ¨ P pbpQq q “ rpqqq . Writing β “

52{3 ¨15 28

and β 1 “

32{3 ¨45 , 141{3 ¨8

by Propositions 3.4.3 and 3.1.4, we thus have

(3.8.5) P pbpQq q “ rpqq, sbpRq q “ sprpqqqq “

βA pβδr pqqq β 1 A pβ 1 δs pqqq p1 ` op1qq . q 2{3 rpqq2{3

Next, |P pbpQq q “ rpqq, sbpRq q “ sprpqqqq ´ P pbpQq q “ rpqq, sbpQq q “ sprpqqqq| ď P pbpQq q “ rpqq, sbpQq q ‰ sprpqqq, sbpRq q “ sprpqqqq (3.8.6)

` P pbpQq q “ rpqq, sbpQq q “ sprpqqq, sbpRq q ‰ sprpqqqq ,

If tbpQq q “ rpqq, sbpQq q ‰ sprpqqq, sbpRq q “ sprpqqqu occurs then L1 pQq q ě sprpqqq, as explained in Fact 3.8.1. Similarly, if tbpQq q “ rpqq, sbpQq q “ sprpqqq, sbpRq q ‰

82

3.8. PROOFS OF THE MAIN THEOREMS

83

sprpqqqu occurs then Qq must contain a simple block of size sprpqqq that does not lie within Rq ; in this case we also obtain L1 pQq q ě sprpqqq. It follows from Proposition 3.5.3 that there exist positive constants c1 , c2 , c3 such that |P pbpQq q “ rpqq, sbpRq q “ sprpqqqq ´ P pbpQq q “ rpqq, sbpQq q “ sprpqqqq | ď 2P pL1 pQq q ě sprpqqqq ď c1 exp p´c2 q c3 q ` ˘ “ o q ´2 , which combined with (3.8.5) proves the theorem.



For the proof of Theorem 3.1.3, we require a lemma bounding the maximum degree in a quadrangulation uniformly drawn from Qq,rpqq,spqq ; the lemma follows easily from the fact that degrees in uniform quadrangulations have exponential tails.

Lemma 3.8.2. Let Qq Pu Qq,rpqq,spqq . Then for all q sufficiently large, ¯ ´ P maxpdegQq pwq : w P vpQq qq ě pln qq2 ă q ´10 . Proof. By [18, Theorem 2.1 (a)] (and Tutte’s angular bijection between maps and quadrangulations), for all ε ą 0 there exists B ą 0 such that for all q ě 3, if Q Pu Qq and u Pu vpQq then ˆ `

(3.8.7)

˘

P degQ puq ą d ă B

1 `ε 2

˙d .

Given that bpQq “ rpqq and sbpQq “ spqq, the conditional law of Q is uniform on Qq,rpqq,spqq ; so ´ ¯ P maxpdegQq pwq : w P vpQq qq ą d ˇ ` ˘ “ P maxpdegQ pwq : w P vpQqq ą d ˇ bpQq “ rpqq, sbpQq “ spqq ˇ ` ˘ ď q ¨ P degQ puq ą d ˇ bpQq “ rpqq, sbpQq “ spqq

3.8. PROOFS OF THE MAIN THEOREMS

84

` ˘ P degQ puq ą d ďq¨ P pbpQq “ rpqq, sbpQq “ spqqq ` ˘ “ Opq 7{3 q ¨ P degQ puq ą d , the final inequality by Theorem 3.1.4 and the definition of rpqq and spqq. Taking d “ ln2 q and ε ă 1{2, the result then follows from (3.8.7).



Proof of Theorem 3.1.3. Recall that Qq Pu Qq,rpqq,spqq , Rq “ RpQq q and Sq “ d

SpQq q. Let Q Pu Qq . Given that bpQq “ rpqq and sbpQq “ spqq, we have Qq “ Q. By Fact 3.8.1, we then have ˇ ` ˘ P pSq ‰ SpRq qq “ P SpQq ‰ SpRpQqqˇbpQq “ rpqq, sbpQq “ spqq ˇ ` ˘ ď P L1 pQq ě spqqˇbpQq “ rpqq, sbpQq “ spqq ˇ ` ˘ P L1 pQq ě spqqˇbpQq “ rpqq ď . P pbpQq “ rpqq, sbpQq “ spqqq Combined with Theorem 3.1.4, this gives ˇ ` ˘ ` ˘ P pSq ‰ SpRq qq “ O q 4{3 ¨ P L1 pQq ě spqqˇbpQq “ rpqq . ` ˘ Since spqq “ q{3 ` O q 2{3 , by Proposition 3.5.3 there exist c2 , c3 ą 0 such that ˇ ` ˘ P L1 pQq ě spqqˇbpQq “ rpqq “ O pexp p´c2 q c3 qq . Hence, (3.8.8)

` ˘ P pSq ‰ SpRq qq “ O q 4{3 ¨ exp p´c2 q c3 q .

Now let R Pu Rrpqq,spqq . Given that Sq “ SpRq q, we have Rq Pu Rrpqq,spqq , so (3.8.8) implies easily that for any bounded continuous function g : K2 Ñ R ˇ „ ˆ´ „ ˆ´ ˙ˇ ¯1{4 ´ ¯1{4 ˙ ¯1{4 ´ ¯1{4 ˇ ˇ 21 21 21 21 ˇE g ˇÑ0, R , S ´ E g R, SpRq q q 40rpqq 40rpqq 40rpqq 40rpqq ˇ ˇ

3.8. PROOFS OF THE MAIN THEOREMS

85

as q Ñ 8. By Proposition 3.1.1 and the Portmanteau theorem, it follows that as rpqq Ñ 8, ˜ˆ (3.8.9)

21 40rpqq

˙1{4

ˆ Rq ,

21 40rpqq

¸

˙1{4 Sq

d

Ñ pm8 , m8 q .

Moreover, by the definition of rpqq, there exist C1 , C2 ą 0 such that for all q ą 0, 9 9 21 ď ď . 2{3 8q ` C1 q 40rpqq 8q ´ C2 q 2{3 From this and (3.8.9) we obtain ˜ˆ ˙ ˆ ˙1{4 ¸ 1{4 9 9 d Rq , Sq Ñ pm8 , m8 q (3.8.10) 8q 8q as q Ñ 8. To finish the proof, we show that also ˜ˆ ˙ ˆ ˙1{4 ¸ 1{4 9 9 d (3.8.11) Qq , Rq Ñ pm8 , m8 q. 8q 8q Joint convergence of the triple to the limit pm8 , m8 , m8 q is immediate from (3.8.10) and (3.8.11), so it remains to prove (3.8.11). (Note that we may not simply invoke ´ ¯1{4 d 9 the result of Le Gall [48] and of Miermont [59] to conclude that the 8q Qq Ñ m8 since Qq is not uniformly distributed over Qq , but over Qq,rpqq,spqq .) The argument is similar to that in Proposition 3.1.1, and we focus on explaining the points where it differs. Let e1 be the ăQq -minimal oriented edge of Rq ; by definition, this is the root edge of Rq . Write Qq “ pQq , eq and Rq “ pRq , e1 q. Also, let Q1q “ pQq , e1 q. The bijection ψ from Proposition 3.1.3 gives a decomposition of Q1q as pRq , ppLi , bi q : 0 ď i ď 2rpqq ´ 4qq , where the Li “ pMi,j : 1 ď j ď `i q P Q`i satisfy (recalling (3.1.2)) (3.8.12)

|epRq q| `i ÿ ÿ

|epQq q| “ |epRq q| ` i“0

p|epMi,j q| ` 1 ` 1r|epMi,j q|‰1s q ,

j“1

3.8. PROOFS OF THE MAIN THEOREMS

86

and bi “ pbi,j : 1 ď j ď `i q P t0, 1u`i . List the elements of epRq q as pei : 1 ď i ď |epRq q|q according to the order ăRq ; like in Section 3.1.1, we view ei as oriented (we oriented so that the tail e´ i precedes the head e` i according to the breadth-first order described in the introduction, but this is unimportant; all that matters is to have a fixed rule for choosing the orientation). Also, let e0 be a copy of e1 . Under the bijection ψ, for each 0 ď i ď |epRq q| and 1 ď j ď `i , the value bi,j indicates the endpoint ei at which Mi,j is attached. Recall that µB “ µB Qq is the degree-biased measure on vpQq q, We now compare µB with a random projection of µB onto Rq . First define a vector nq as follows. Let ř0 n0 “ 0 if `0 “ 0 and otherwise let n0 “ `j“1 p|epM0,j q| ` 1 ` 1r|epM0,j |‰1s q, and for 1 ď i ď 2rpqq ´ 4 let (3.8.13)

ni “ 1 `

`i ÿ

p|epMi,j q| ` 1 ` 1r|epMi,j q|‰1s q

j“1

Set nq “ pni : 0 ď i ď 2rpqq ´ 4q; it is immediate from Proposition 3.1.3 that nq n

is exchangeable. Now define the measure ν nq “ νRqq as in (3.6.2): more precisely, for each edge ei P epRq q choose a uniformly random endpoint wi of ei . Then ν nq is specified by letting ν nq pV q “

ÿ 1 ni 2q ´ 4 ti:w PV u i

for V Ă vpRq q. (The fact that 2q ´ 4 is the correct normalizing constant follows from (3.8.12).) p

p

If maxpni : 0 ď i ď 2rpqq ´ 4q{p2q ´ 4q Ñ 0 then |nq |2 {|nq |1 Ñ 0 and the same p

argument which led to Corollary 3.7.2 gives dP pµRq , ν nq q Ñ 0. Assuming this holds then just as in (3.7.5) we obtain (3.8.14)

´ ´ ¯¯ p 9 1{4 9 1{4 q Rq , vpRq q, p 8q q ¨ dRq , ν nq dGHP p 8q Ñ 0.

3.9. TUTTE’S BIJECTION AND THE QUADRATIC METHOD

87

Recall the definition of D1 from (3.5.8). Reprising the argument for (3.7.8) now gives that for ε ą 0, (3.8.15)

dGHP ppvpRq q, εdRq , ν nq q, εQq q ď 2ε ¨ pD1 pQq q ` 1q ` maxpε, 1{qq .

This has a very slightly different form from (3.7.8), where the bound was 2εDpRr q ` maxpε, 1{rq. The reason for the difference is that in the current setting, the submaps of Qq pendant to Rq only attach to one end of an edge of Rq . When we project the mass to form ν nq we may choose the “wrong end”. This source of error did not appear when projecting mass onto the largest simple block because the 2-connected “decorations” of the largest simple block are attached at both endpoints of their respective edges. p

At any rate, by Proposition 3.5.4, D1 pQq q{q 1{4 Ñ 0, so (3.8.15) and (3.8.14) p

9 1{4 9 1{4 together give dGHP pp 8q q Qq , p 8q q Rq q Ñ 0. But by (3.8.10) we know that the

second argument converges to m8 , and (3.8.11) follows. p

It thus remains to prove that maxpni : 0 ď i ď 2rpqq ´ 4q{p2q ´ 4q Ñ 0. But this is easy: `i is the number of copies of a particular edge in Qq , so max0ďiď2q´4 `i is at most maxpdegQq pwq : w P vpQq qq. By (3.8.13) we then have maxpni : 0 ď i ď 2rpqq ´ 4q ď 1 ` maxpdegQq pwq : w P vpQq qq ¨ p2 ` max |epMi,j q|q. i,j

By Lemma 3.8.2, the largest degree is at most ln2 q with high probability, and p

Proposition 3.5.3 gives that q ´3{4 ¨ maxi,j |epMi,j q| ď q ´3{4 ¨ p2L1 pQq q ´ 4q Ñ 0. The result follows.



Proof of Theorem 3.1.2. The theorem follows from Theorem 3.1.3 in exactly the same way as Theorem 3.1.1 followed from Proposition 3.1.1, using Theorem 3.1.4 in place of Proposition 3.1.4 for the averaging argument.



3.9. Tutte’s Bijection and the Quadratic Method 3.9.1. Tutte’s Bijection. Given a rooted map pM, eq, we call the face of M to the right of e its root face, recalling that e is an oriented edge. By Tutte’s bijection,

3.9. TUTTE’S BIJECTION AND THE QUADRATIC METHOD

pM, eq can be uniquely transformed into a rooted quadrangulation pQ, e1 q in the following way: (1) In the interior of each face f of M , insert a new vertex v. In particular, we call the vertex inserted in the root face of pM, eq the outer vertex. (2) In each face f of M , add an edge between the newly inserted vertex v and each vertex of M which is incident to f . (3) Let vertex set vpQq be the union of vpM q and the set of newly inserted vertices. Let edge set epQq be the set of newly created edges. Let e1 be the edge of Q which succeeds e in the clockwise order around the tail te of e, oriented from te to the outer vertex of Q. This procedure can be reversed uniquely, so there is a bijection between the set of rooted maps with n edges and the set of rooted quadrangulations with n faces. 3.9.2. The Quadratic Method. Proof of Lemma 3.4.4. Let mpz, uq be the generating function of rooted maps, where z marks the number of edges, and u marks the number of edges incident to the root face, called the outer edges. Define qpz, uq “

z 2 u4 ´ 2zu2 pu ´ 1qp2u ´ 1q ` p1 ´ u2 q . 4z 2 u2 p1 ´ uq2

By Tutte’s enumeration method (see [36, VII 8.2]), mpz, uq and qpz, uq satisfy ˆ (3.9.1)

1 ´ u ` zu2 mpz, uq ´ 2zu2 p1 ´ uq

˙2 “ qpz, uq `

mpz, 1q . up1 ´ uq

Next, let M pz, uq be the generating function of rooted quadrangulations, where z marks the number of edges, and u marks the number of faces incident to the outer vertex. Note that for any i, j P N, a rooted map of i edges and j outer edges is uniquely identified with a rooted quadrangulation of i faces and j faces incident to the outer vertex. It follows from Euler’s formula that a quadrangulation of i faces has 2i edges. So M pz, uq “ mpz 2 , uq .

88

3.10. REMAINING DERIVATION USING SINGULARITY ANALYSIS

89

Define Qpz, uq “ qpz 2 , uq “

(3.9.2)

z 4 u4 ´ 2z 2 u2 pu ´ 1qp2u ´ 1q ` p1 ´ u2 q . 4z 4 u2 p1 ´ uq2

Then by change of variable in (3.9.1), M pz, uq and Qpz, uq satisfy ˆ (3.9.3)

1 ´ u ` z 2 u2 M pz, uq ´ 2 2 2z u p1 ´ uq

˙2 “ Qpz, uq `

M pz, 1q . up1 ´ uq

We want to determine the implicit function u “ upzq so that both sides of the equation (3.9.3) is equal to 0. Notice that we have double roots in u, so (3.9.4)

Qpz, uq `

M pz, 1q “0, up1 ´ uq

2u ´ 1 B Qpz, uq ` 2 M pz, 1q “ 0 . Bu u p1 ´ uq2 By substituting for M pz, 1q we get

B Qpz, uq Bu

´

2u´1 Qpz, uq up1´uq

“ 0. It then follows

from (3.9.2) that pu2 z 2 ` u ´ 1qru2 z 2 ` pu ´ 1qp2u ´ 3qs “ 0 . Suppose the first term is equal to 0, then it follows from (3.9.2) and (3.9.4) that M pz, 1q “ up4´3uq . p2u´3q2

1 , z2

which is not true. Thus, z 2 “

Letting t “ 1 ´

1 u

p1´uqp2u´3q , u2

and so M pzq “ M pz, 1q “

yields ˆ

t“z

t 1 ´ 3t

˙1{2 , M pzq “

1 ´ 4t . p1 ´ 3tq2

t q1{2 , and let LM pzq be the implicit function Furthermore, define φM ptq “ p 1´3t

that solves LM pzq “ zφM pLM pzqq. Finally, setting ψM ptq “

1´4t p1´3tq2

gives M pzq “

ψM pLM pzqq.



3.10. Remaining Derivation Using Singularity Analysis We present the remaining derivation for the expressions in Tables 3.1 and 3.2.

3.10. REMAINING DERIVATION USING SINGULARITY ANALYSIS

90

We first establish a system of Lagrangean equations for the generating function Cpzq. Then we derive the expansion of a Lagrangean series LC pzq around its singularity, where LC pzq is implicitly defined for Cpzq. Furthermore, we express the generating function Cpzq in terms of LC pzq to obtain a singular expansion of Cpzq. Lemma 3.10.1. Let ψC ptq “

t2 p´1`5tq , p´1`3tq3

2 2

q let φC ptq “ ´ p1´5t`8t , and let LC pzq be p´1`3tq3

defined by the implicit relation LC pzq “ zφC pLC pzqq, then Cpzq “ ψC pLC pzqq . Proof. Write h “ Hpzq, and view z “ zphq as a function of h. In Lemma 3.4.6, 3

tp´1`3tq we show that taking ψH ptq “ ´ p1´5t`8t 2 q2 yields Hpzq “ ψH pLM pzqq. It follows that,

setting t “ LM pzq gives h “ ψH pLM pzqq “ ´

tp´1 ` 3tq3 . p1 ´ 5t ` 8t2 q2

Let LC phq “ LM pzq “ t, then LC phq p1 ´ 5t ` 8t2 q2 “´ . h p´1 ` 3tq3 2 2

q Setting φC ptq “ ´ p1´5t`8t then yields LC phq “ hφC pLC phqq. p´1`3tq3

Finally, by (3.3.6) Cphq “ CpHpzqq “ M pzq ´ 2zp1 ` M pzqq2 . We may write M pzq “ ψM pLM pzqq “ ψM ptq, and z “ Cphq “

t2 p´1`5tq . p´1`3tq3

Hence, setting ψC ptq “

t2 p´1`5tq p´1`3tq3

In the remainder of this chapter, write ρ “



t . φM ptq

gives Cphq “ ψC pLC phqq.

27 . 196

2

, then Lemma 3.10.2. Let ψU ptq “ ´ tp´1`4tq p´1`3tq3 U pzq “ ψU pLC pzqq , where LC pzq is given in Lemma 3.10.1.

LM pzq φM pLM pzqq

Then 

3.10. REMAINING DERIVATION USING SINGULARITY ANALYSIS

91

Proof. Let φC ptq , ψC ptq be given in Lemma 3.10.1. By the definition of U given in (3.3.4), we have U pzq “ z p1 ` Cpzqq2 “

LC pzq ¨ p1 ` ψC pLC pzqqq2 . φC pLC pzqq

2

Setting ψU ptq “ ´ tp´1`4tq gives U pzq “ ψU pLC pzqq. p´1`3tq3 Write LC pzq “ τ ´ d1 p1 ´ z{ρq1{2 ` O p1 ´ z{ρq as its singular expansion (derived similarly as for (3.4.3)). Using Taylor’s expansion then gives ´ ¯ U pzq “ ψU pLC pzqq “ ψU τ ´ d1 p1 ´ z{ρq1{2 ` O p1 ´ z{ρq ´ ¯2 1 2 1{2 “ ψU pτ q ` ψU pτ q ´d1 p1 ´ z{ρq 2 ´ ¯3 ` ˘ 1 3 ` ψU pτ q ´d1 p1 ´ z{ρq1{2 ` O p1 ´ z{ρq2 6 ` ˘ “ ψU pτ q ´ u1 p1 ´ z{ρq ` u3{2 p1 ´ z{ρq3{2 ` O p1 ´ z{ρq2 , where ψU pτ q “

4 , 27

u1 and u3{2 are as displayed in Table 3.2. Note that U pρq “

ψU pτ q. The parameterization for B can be similarly derived, so we omit the details.



The Brownian Plane with Minimal Neck Baby Universe

93

CHAPTER 4

The Brownian Plane with Minimal Neck Baby Universe 4.1. Introduction The scaling limit of large random planar maps has been a focal point of probability research in the recent decade. Le Gall [48] and Miermont [59] independently established that the Brownian map is the scaling limit of several important families of planar maps; Bettinelli, Jacob & Miermont [23] and Abraham [1], respectively, proved that general and bipartite planar maps with a fixed number of edges converge to the Brownian map after rescaling; Addario-Berry & Albenque [4] and AddarioBerry & Wen [5], respectively, showed that simple quadrangulations and 2-connected quadrangulations also rescale to the same limit object. Curien & Le Gall [30] defined an infinite-volume version of the Brownian map, called the Brownian plane. It shares numerous similarities with the Brownian map, but additionally possesses the scaling invariance property. In this chapter, we describe a pointed measured metric space S obtained by identifying a random point of the Brownian map and the distinguished point of the Brownian plane; a formal definition appears in Section 4.10. This random geometry structure provides a probabilistic model of the so-called minimal neck baby universe in two-dimensional quantum gravity; see Jain & Mathur [40]. Motivated by this notion, we call S the Brownian plane with minimal neck baby universe (minbus in the literature). We show that S is the limit of rescaled uniform quadrangulations conditioned on having an exceptionally large root block.

4.1.1. Notation. A cycle C in a map M is facial if at least one connected component of S2 zC contains neither vertices nor edges of M . Write M ˝ for the map obtained from M by collapsing each facial 2-cycle into an edge. (Notice the difference

4.1. INTRODUCTION

94

from the previous chapter, where M ˝ denotes the map obtained from collapsing each nearly facial 2-cycle.) Fix a rooted quadrangulation Q “ pQ, uvq. Given an edge e P epQq, let Be Ă vpQq be maximal subject to the constraints that QrBe s is 2-connected and that both endpoints of e are in Be ; we call QrBe s˝ a block in Q. (In this chapter, when we say block we mean 2-connected block.) In particular, we call pQrBuv s, uvq the preroot-block of Q, and call RpQq :“ QrBuv s˝ the root block of Q. (Notice that in the previous chapter, we used R‚ pQq to denote the root block.) Now we briefly describe submaps pendant to the pre-root-block, referring to Sections 4.3.1 and 4.3.2 for an elaboration. We write F “ FpQq “ tf p1q, . . . , f p|F|qu for the set of facial 2-cycles in the pre-root-block of Q. For i P t1, . . . , |F|u, write Pi “ Pi pQq for the unique maximal connected submap of Q lying in the face enclosed by f piq and containing no edge of f piq. Next, let Λ “ ΛpQq P tP1 , . . . , P|F | u be the element with the largest size, assuming that Λ is non-empty; this condition is always satisfied for the quadrangulations we consider in the sequel. Recall from Section 2.1.4 that ăQ is a total order on the edge set of Q. If there are multiple elements of tP1 , . . . , P|F | u of maximal size, we take Λ to be the one which contains the ăQ -smallest edge. Furthermore, write ρQ “ vpRpQqq X vpΛpQqq , and let R` pQq “ Q ´ v pΛpQqq ztρQ u . Recall that we write Q and R for the sets of connected and 2-connected rooted quadrangulations, respectively. For all r P N with r ă n, let Qrn “ tQ P Qn : |vpRpQqq| “ ru .

4.1. INTRODUCTION

4.1.2. Convergence in the Local Gromov-Hausdorff-Prokhorov Topology. For the current subsection, a reference to Appendix 4.10 for the definition of S may be helpful, though the intuition of S presented above should be sufficient for the comprehension of the following contents. Since S contains the Brownian plane as a subspace, S is non-compact and the usual Gromov-Hausdorff-Prokhorov (GHP) topology is too strong for convergence, so we need the notion of local GHP topology. We now briefly describe this topology; for further details, see Abraham, Delmas & Hoscheit [2, Section 2], Burago, Burago & Ivanov [26, Chapter 8], Curien & Le Gall [30, Section 2.1], or Section 4.2.2 of this chapter. We call a metric space pV, dq a length space if, for any x, y P V , dpx, yq equals the infimum of the lengths of continuous curves connecting x and y. We call a metric space pV, dq boundedly compact if all closed balls of finite radius are compact. A pointed measured metric space is a quadruple pV, d, o, νq, where pV, dq is a metric space, o P V , and ν is a non-negative finite Borel measure on pV, dq. Given a pointed measured metric space V “ pV, d, o, νq, for any r ě 0, let ´ ˇ ¯ Br “ Br pVq “ tw P V : dpw, oq ď ru, and write Br pVq “ Br , d, o, ν ˇBr ; here and in the sequel, we often abuse notation and use d to denote the metric restricted to a subspace. Informally, a sequence of pointed measured boundedly compact length spaces pVn : n P Nq converges to V in the local GHP topology if for any r ě 0, Br pVn q converges to Br pVq in the pointed GHP topology; see Section 4.2.1 for the definition of pointed GHP topology. A rooted graph pG, uvq is not a length space, but we may approximate it by a pointed boundedly compact length space so that the balls centred at u in pG, uvq and at the distinguished point in the approximating space are within GHP distance 1. Throughout the chapter, when we say that graphs converge in the local or pointed GHP topology, we mean for their approximating spaces. More precisely, given a graph G, we view each edge of G as an isometric copy of the unit interval r0, 1s. Abusing notation, we continue to write G for the resulting length space, and let dG be the intrinsic metric; see [26, Chapter 2] for details on length spaces. Finally, we

95

4.1. INTRODUCTION

96

let νG “

ÿ

δv

vPvpGq

be the counting measure on vpGq. Theorem 4.1.1 Let r : N Ñ N be such that rpnq ą pln nq25 for all n and rpnq “ opnq as ´ ¯1{4 rpnq 40¨rpnq n Ñ 8. Then for Qn Pu Qn , writing kn “ , we have 21 ˆ

8 1 1 ¨ dQn , ρQn , ¨ νR` pQn q Qn , ¨ νΛpQn q ` 4 ` kn 9kn |vpR pQn qq|

˙ ÑS

in distribution for the local Gromov-Hausdorff-Prokhorov topology. By assigning 0 mass to components of R` pQn q ´ vpRpQn qq, we obtain a similar scaling limit result. Theorem 4.1.2 Let r : N Ñ N be such that rpnq ą pln nq25 for all n and rpnq “ opnq as ¯1{4 ´ rpnq n Ñ 8. Then for Qn Pu Qn , writing kn “ 40¨rpnq , we have 21 ˆ

8 1 1 ¨ dQn , ρQn , Qn , ¨ νΛpQn q ` ¨ νRpQn q 4 kn 9kn rpnq

˙ ÑS

in distribution for the local Gromov-Hausdorff-Prokhorov topology. The proof of Theorem 4.1.2 is similar to but simpler than that of Theorem 4.1.1, so we only provide a proof outline for Theorem 4.1.1. In the remainder of the chapter, let r : N Ñ N be such that rpnq ą pln nq25 for all n and rpnq “ opnq as n Ñ 8; the first assumption is necessary due to (4.7.3), while the second one allows us to obtain a non-compact metric space in the limit. For all n P N, let ˆ kn “

40 ¨ rpnq 21

˙1{4 .

Whenever we refer to [5], readers may also find the same contents in the previous chapter.

4.1. INTRODUCTION

97

4.1.3. Proof Outline for Theorem 4.1.1. Write P for the pointed measured Brownian plane and m8 for the pointed measured Brownian map, both endowed with uniform measures; see Sections 1.1.3 and 1.2.1 for the definitions. rpnq In this subsection, for all n P N, let Qn Pu Qn . It is easily seen that Rn :“

RpQn q is a uniform 2-connected quadrangulation with rpnq vertices. Then it follows from [5, Theorem 1.1] (Theorem 3.1.1) that ˆ p n :“ R

1 1 Rn , ¨ dRn , ρQn , ¨ νRn kn rpnq

˙ d

Ñ m8

as n Ñ 8 for the pointed GHP topology. Write Rn` “ Rn` pQn q. To establish an analogous convergence result for Rn` , we show that components of Rn` ´vpRn q are uniformly asymptotically negligible. This is accomplished in two main steps. First, Proposition 4.5.2 and Corollary 4.7.2 prove that each component is small in size. Then, using the quartic relation between size and diameter, Corollary 4.7.4 shows that the diameters of these components have order oprpnq1{4 q with high probability, proving their negligibility in terms of metric structure. Secondly, Lemmas 4.8.3 and 4.8.4 show that these components do not concentrate on a small region, proving their negligibility in terms of measure structure. Then it follows that ˆ (4.1.1)

p ` :“ R n

Rn` ,

1 1 ¨ dRn` , ρQn , ¨ν ` kn |vpRn` q| Rn

˙ d

Ñ m8

for the pointed GHP topology, as shown in Proposition 4.8.2. ¯ ´ On the other hand, we prove that Λn :“ ΛpQn q has Ω plnnnq2 vertices with high probability; details appear in Proposition 4.5.1 and Corollary 4.7.1. Note that conditioned on its size, Λn is a uniform quadrangulation. Then it follows from [30, Theorem 2] (Theorem 1.2.2) that ˆ (4.1.2)

p n :“ Λ

1 8 Λn , ¨ dΛn , ρQn , ¨ νΛ n kn 9kn4

˙ d

ÑP

4.2. POINTED AND LOCAL GROMOV-HAUSDORFF-PROKHOROV DISTANCES 98

for the local GHP topology; the convergence in [30] is stated for the local GromovHausdorff topology, but a slight extension of their proof in fact yields the above formulation, as shown in Section 4.11.2. By (4.1.1) and (4.1.2) we easily obtain the joint convergence (4.1.3)

´ ¯ d p `, Λ pn Ñ R pm8 , Pq n

for the local GHP topology, where m8 and P are independent, as explained in Lemma 4.9.2. Finally, we view Qn as a space obtained by gluing Rn` to Λn at the point ρQn , and analogously view S as m8 glued to P. Lemma 4.9.1 shows that local GHP convergence is preserved by such a gluing operation. Theorem 4.1.1 then follows easily from (4.1.3), as shown near the end of Section 4.9. 4.1.4. Organization of the Chapter. Relevant definitions for GHP topologies are given in Section 4.2. We associate quadrangulations to a balls-in-boxes model in Section 4.3, and describe an asymptotically stable distribution for sizes of pendant submaps in Section 4.4. Then we deduce asymptotics for occupancy in a random allocation model with a varying balls-to-boxes ratio, given in Section 4.5. In Section 4.6, we derive a bound for the number of pendant submaps of the root block. Size-bounds and diameter-bounds for quadrangulations are shown in Section 4.7. Then we establish (4.1.1) in Section 4.8, by showing that uniformly asymptotically negligible attachments do not affect the scaling limit. We complete the proofs for Theorems 4.1.1 and 4.1.2 in Section 4.9. In Section 4.10, we present a formal definition for the Brownian plane with minbus. Finally, in Section 4.11, we present an extension of the convergence result from [30] to the local GHP topology, following a review of the scaled Brownian map. 4.2. Pointed and Local Gromov-Hausdorff-Prokhorov Distances In this section, we briefly recall the definitions of pointed and local GromovHausdorff-Prokhorov distances.

4.3. MAP DECOMPOSITION AND BALLS-IN-BOXES MODEL

4.2.1. Pointed Gromov-Hausdorff-Prokhorov Distance. Recall the definition of pointed measured metric space from Section 4.1.2. The pointed GromovHausdorff-Prokhorov distance between two pointed measured metric spaces V “ pV, d, v, µq and W “ pW, d1 , w, µ1 q is d‹GHP pV, Wq “ inf rmax tδH pφpV q, φ1 pW qq, δP pφ˚ µ, φ1˚ µ1 q, δpφpvq, φ1 pwqqus , where the infimum is taken over all isometries φ and φ1 from pV, dq and pW, d1 q, respectively, into a common metric space pZ, δq. Pointed measured metric spaces pV, d, v, µq and pW, d1 , w, µ1 q are isometry-equivalent if there exists a measurable bijective isometry Φ : V Ñ W such that Φ˚ µ “ µ1 and Φpvq “ w. Write K‹ for the set of isometry-equivalence classes of pointed measured compact metric spaces, then pK‹ , d‹GHP q is a Polish space; see [4]. 4.2.2. Local Gromov-Hausdorff-Prokhorov Distance. We quickly review the definition of local GHP distance, and refer the reader to [30, Section 1.2] or [26, Section 8.1] for further explanations. The local Gromov-Hausdorff-Prokhorov distance between two pointed measured metric spaces V “ pV, d, v, µq and W “ pW, d1 , w, µ1 q is 8 ÿ min td‹GHP pBr pVq, Br pWqq , 1u dLGHP pV, Wq “ . 2r r“1

Recall the definition of boundedly compact length space from Section 4.1.2. Write KL for the set of isometry-equivalence classes of pointed measured boundedly compact length spaces. A recent result [2, Theorem 2.9] shows that pKL , dLGHP q is also a Polish space. Throughout the chapter, when we say convergence for the local (resp. pointed) GHP topology, we mean convergence in the space pKL , dLGHP q (resp. pK‹ , d‹GHP q). 4.3. Map Decomposition and Balls-in-Boxes Model Fix Q P Q. Recall from Section 4.1.1 that F “ FpQq denotes the set of facial 2-cycles in the pre-root-block of Q, and that pPi pQq : 1 ď i ď |F|q lists the submaps

99

4.3. MAP DECOMPOSITION AND BALLS-IN-BOXES MODEL

enclosed by the facial 2-cycles. When Q is random, we are able to recast the behaviour of p|vpPi pQqq| : 1 ď i ď |F|q as a balls-in-boxes allocation problem with unlabelled balls (corresponding to the submaps) and labelled boxes (corresponding to the facial 2-cycles). In this section, we present a map decomposition which we use to elaborate this viewpoint. We emphasize that the map decomposition here is different from the block decomposition given by Addario-Berry [3].

4.3.1. Pendant Submap. Fix a rooted map M “ pM, uvq in this subsection. A corner of M incident to a vertex v P vpM q is an ordered pair pe, e1 q such that te, e1 u Ă epM q, e and e1 are incident to v, and e1 follows e immediately in the clockwise order around v. (If v has degree 1 then e “ e1 .) Let CpM q be the set of corners of M . Given c P CpM q, write vpc, M q and f pc, M q for the vertex and face incident to the corner c in M , respectively. It often causes no confusion to write vpcq in place of vpc, M q.

f pc, Rq

e

ec

vpcq c (a) a rooted map M

e1

(b) a submap R of M

Figure 4.1. M is a rooted map with black and red vertices. R is the submap of M induced by the black vertices. Also, c is a corner of R, and f pc, Rq is the face of R incident to c (the shaded area). Finally, Vc pM, Rq are red vertices.

For this paragraph, a reference to Figure 4.1 may be helpful. Given a connected submap R of M , for each c P CpRq, let Vc pM, Rq “ tu P vpM q : D a path in M from u to vpcq lying within f pc, Rq and disjoint from vpRqztvpcquu .

100

4.3. MAP DECOMPOSITION AND BALLS-IN-BOXES MODEL

(It is possible that Vc pM, Rq “ H.) Let Pc1 pM, Rq “ M rVc pM, Rq Y tvpcqus. Write ec for the ăM -minimal edge in Pc1 pM, Rq; or if Vc pM, Rq “ H then let ec “ tvpcqu. Finally, let P1c pM, Rq “ pPc1 pM, Rq, ec q be the rooted submap.

(a) rooted quadrangulation Q

cp1q cp3q

f p1q

f p3q cp2q cp4q

cp5q cp7q

f p2q

cp6q cp8q f p4q

(b) pre-root-block B of Q

P1cp1q P1cp2q P1cp3q P1cp4q P1cp5q P1cp6q P1cp7q P1cp8q (c) the sequence of submaps pP1cpiq : 1 ď i ď 2|F|q pendant to B in Q

Figure 4.2.

101

4.3. MAP DECOMPOSITION AND BALLS-IN-BOXES MODEL

4.3.2. Decomposition and Allocation. In this subsection, fix n P N and rpnq

Q “ pQ, uvq P Qn . For the current paragraph, it is convenient to keep Figure 4.2 on hand. Let B be the pre-root-block of Q, recalling that B is rooted. We require a canonical ordering of the corners of facial 2-cycles in B. Its precise form is unimportant but we describe it nonetheless for completeness. Recall from Section 2.1.4 that ăB and ăB are the total orders on the vertex set and the edge set of B, respectively. List the vertices of B as u1 “ u, u2 “ v, u3 , ..., urpnq in the ăB -order. Write F “ FpQq for the set of facial 2-cycles, and list them as f p1q, ..., f p|F|q in the ăB -order of their incident edges ui uj with i ă j, and use the clockwise order around ui to determine priority; we call this the canonical order of B. For each 1 ď i ď |F|, let cp2i ´ 1q and cp2iq be the corners incident to f piq, where cp2i ´ 1q is incident to the vertex with smaller index. Given 1 ď i ď 2|F|, let P1cpiq “ P1cpiq pQ, Bq be the pendant submap of Q incident to cpiq. Note that half of the time P1cpiq is a single vertex, so we may list the non-trivial ´ ¯ 1 1 elements of the sequence Pcpiq : |vpPcpiq q| ě 2, 1 ď i ď 2|F| as (4.3.1)

pPi pQq : 1 ď i ď |F|q .

(Pi pQq is the rooted version of Pi pQq as introduced in Section 4.1.1.) Notice that pP1cpiq : |vpP1cpiq q| ě 2, 1 ď i ď 2|F|q and pPi pQq : 1 ď i ď |F|q are in the same order. ¯ ´ 1 It follows that Pcpiq : 1 ď i ď 2|F| may be recovered from the pair of the sequences ´ ¯ pPi pQq : 1 ď i ď |F|q and 1r|vpP1cpiq q|ě2s : 1 ď i ď 2|F| . Next, given integers m ě 1 and k ě 1, write # Bm,k “

py1 , ¨ ¨ ¨ , yk q P Nk :

k ÿ

+ yi “ m

i“1

for the set of possible allocations of m (unlabelled) balls in k (labelled) boxes. For each 1 ď i ď |F|, let (4.3.2)

Yi pQq “ |vpPi pQqq| ´ 1 .

102

4.4. ASYMPTOTICALLY STABLE DISTRIBUTION

103

Since |vpPi pQqq| ě 2 by definition, we have Yi pQq ě 1. It is easily seen that |F | ÿ

Yi pQq “ n ´ rpnq ,

i“1

and pYi pQq : 1 ď i ď |F|q P Bn´rpnq,|F | . We call pYi pQq : 1 ď i ď |F|q the allocation associated with Q. Conversely, for a given allocation pyi : 1 ď i ď |F|q P Bn´rpnq,|F | , there are rpnq

multiple rooted quadrangulations Q1 P Qn

such that |vpPi pQ1 qq| ´ 1 “ yi for each

1 ď i ď |F|. We call these the quadrangulations associated with pyi : 1 ď i ď |F|q P Bn´rpnq,|F | ; the number of such quadrangulations Q1 is given in Lemma 4.4.2, below. 4.4. Asymptotically Stable Distribution We use that there exists ψ : N Ñ R with ψpxq Ñ 0 as x Ñ 8 and with ψpxq ą ´1 for all x P N, such that for all integer k ě 2, (4.4.1)

2 12k´2 p1 ` ψpk ´ 1qq |Qk | “ ? ; π pk ´ 1q5{2

see [5, Proposition 3.1 and Corollary 4.5] or [16, Proposition 4]. (There are

12k p1`op1qq ?2 π k5{2

rooted maps with k edges, so by Tutte’s bijection and Euler’s formula there are 12k p1`op1qq ?2 π k5{2

rooted quadrangulations with k faces, or with k `2 vertices, as k Ñ 8.)

Next, for each k P N, let (4.4.2)

wpkq “

1 ` ψpkq wpkq , ppkq “ ř8 . 5{2 k k“1 wpkq

Since pppkq : k P Nq is a probability distribution, we may associate it with a random variable ξ such that P pξ “ kq “ ppkq for all k P N. Let ξ1 , ξ2 , ¨ ¨ ¨ be independent ř copies of ξ. For any k P N, write Sk “ ki“1 ξi . For n, r P N and k P Ně0 with n ě r and k ď n ´ r, let (4.4.3)

r Qr,k n “ tQ P Qn : |FpQq| “ ku ,

recalling that |FpQq| is the number of facial 2-cycles in the pre-root-block of Q.

4.4. ASYMPTOTICALLY STABLE DISTRIBUTION

104

Proposition 4.4.1. Fix n, r, N P N with 1 ď N ď n ´ r, and let pyi : 1 ď i ď N q P Bn´r,N . Then for Qn Pu Qr,N n , ˇ ´ ¯ ˇ P pYi pQn q “ yi , 1 ď i ď N q “ P ξi “ yi , 1 ď i ď N ˇ SN “ n ´ r . The proof of the proposition is given near the end of this section. For any y “ pyi : 1 ď i ď N q P Bn´r,N , let Λyn,r,N be the number of quadrangulations Qn in associated with y, i.e., such that Yi pQn q “ yi for each 1 ď i ď N . To establish Qr,N n the proposition, we first derive an expression for Λyn,r,N .

Lemma 4.4.2. Fix n, r, N P N with 1 ď N ď n ´ r. Then for y “ pyi : 1 ď i ď N q P Bn´r,N , ˙ N ź N ` 2r ´ 4 N |Qyi `1 | . ¨2 “ |Rr | ¨ N i“1 ˆ

Λyn,r,N

associated with the allocation Proof. To build a quadrangulation Q P Qr,N n y “ pyi : 1 ď i ď N q P Bn´r,N , proceed as follows. (1) Let R P Rr . Endow each edge of R with an orientation so that the tail precedes the head in breadth-first order. List the resulting oriented edges as pei : 1 ď i ď |epRq|q in the increasing order of ăR . Let e0 be a copy of e1 lying to the left of e1 , using which we can locate the root edge among multiple edges; see [5, Proposition 1.7] for details. (2) Choose a vector pmi : 0 ď i ď |epRq|q P N|epRq|`1 with

ř2r´4 i“0

mi “ |epRq| `

1 ` N “ 2r ´ 4 ` 1 ` N . Then for each 0 ď i ď |epRq|, split ei into mi copies (if mi “ 1 then there is no split), resulting in N ` 1 facial 2-cycles. Collapse the 2-cycle formed by the rightmost copy of e0 and the leftmost copy of e1 , and root the map at the resulting edge. List the N remaining facial 2-cycles as f p1q, . . . , f pN q in the canonical order described in Section 4.3.2. (3) For each 1 ď i ď N let Qi “ pQi , ui vi q P Qyi `1 . (4) For each 1 ď i ď N , choose one of the two resulting corners incident to f piq and denote it cpiq. Attach Qi to cpiq by identifying vpcpiqq with ui , then

4.4. ASYMPTOTICALLY STABLE DISTRIBUTION

105

add another edge with endpoints ui and vi , drawn so as to quadrangulate the face f piq. In step (1), the number of choices for R is equal to |Rr |. In step (2), the number ř of sequences pmi P N : 0 ď i ď 2r ´ 4q with 2r´4 i“0 mi “ 2r ´ 4 ` 1 ` N is equal `N `2r´4˘ śN to . The number of choices in step (3) is i“1 |Qyi `1 |. In step (4), for N each 1 ď i ď N , there are two ways to choose cpiq, so the total number of choices is 2N . The proof is then concluded by multiplying the previous four numbers of choices.



Corollary 4.4.3. Fix n, r, N P N with 1 ď N ď n ´ r. Then for y “ pyi : 1 ď i ď N q P Bn´r,N , ˙ ˆ ˙N N ź N ` 2r ´ 4 1 1 ` ψpyi q n´r ? “ |Rr | ¨ ¨ 12 . 5{2 N 3 π yi i“1 ˆ

Λyn,r,N

Proof. By (4.4.1), for each i “ 1, . . . , N , 2 12yi ´1 p1 ` ψpyi qq |Qyi `1 | “ ? . 5{2 π yi So N ź

ˆ |Qyi `1 | “

i“1

where

řN

i“1 pyi

2 ? π

˙N řN

12

i“1 pyi ´1q

N ź 1 ` ψpyi q 5{2

i“1

,

yi

´ 1q “ n ´ r ´ N . Then by Lemma 4.4.2, ˙ N ź N ` 2r ´ 4 “ |Rr | ¨ ¨ 2N |Qyi `1 | N i“1 ˆ ˙ ˆ ˙N N ź N ` 2r ´ 4 2 1 ` ψpyi q N n´r´N ? “ |Rr | ¨ ¨2 12 5{2 N π yi i“1 ˆ ˙ ˆ ˙N N ź N ` 2r ´ 4 1 1 ` ψpyi q ? “ |Rr | ¨ ¨ 12n´r . 5{2 N 3 π y i“1 i ˆ

Λyn,r,N



Corollary 4.4.4. Fix n, r, N P N with 1 ď N ď n ´ r, and let pyi : 1 ď i ď N q P Bn´r,N . Then for Qn Pu Qr,N n , P pYi pQn q “ yi , 1 ď i ď N q “ Z ´1

N ź 1 ` ψpyi q 5{2

i“1

yi

,

4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

106

where ÿ

Z“

N ź 1 ` ψpzi q

.

5{2

zi

pz1 ,...,zN qPBn´r,N i“1

associated with y “ pyi : 1 ď i ď N q P Bn´r,N , Proof. For any Q P Qr,N n we have Yi pQq “ yi for each 1 ď i ď N . Then for Qn Pu Qr,N n , it follows from Corollary 4.4.3 that Λy ř n,r,Nz . Λn,r,N

P pYi pQn q “ yi , 1 ď i ď N q “

zPBn´r,N

The corollary follows.



Proof of Proposition 4.4.1. Recalling the definition of ppkq from (4.4.2), it follows from Corollary 4.4.4 that P pYi pQn q “ yi , 1 ď i ď N q “ Z ´1

N ź

wpyi q

i“1

˜ “ Z ´1

(4.4.4)

8 ÿ

k“1

¸N wpkq

N ź

ppyi q .

i“1

On the other hand, since ξ1 , ¨ ¨ ¨ , ξN are independent and that

řN i“1

yi “ n ´ r, we

have ˇ ´ ¯ P pξ “ y , 1 ď i ď N q ˇ i i P ξi “ yi , 1 ď i ď N ˇ SN “ n ´ r “ P pSN “ n ´ rq “ P pSN “ n ´ rq´1

(4.4.5)

N ź

ppyi q .

i“1

The proposition follows immediately by comparing (4.4.4) and (4.4.5).



4.5. Random Allocation with Varying Balls-to-Boxes Ratio Recall from Section 4.1.3 that r : N Ñ N is a function with rpnq ą pln nq25 for all n and rpnq “ opnq. In the remainder of the chapter, for each n P N write mpnq “ n ´ rpnq, and let N : N Ñ N be such that 1 ď N pnq ď mpnq; N pnq corresponds to the number of facial 2-cycles in the pre-root-block of a random quadrangulation with n vertices.

4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

Also recall from Section 4.4 that for k P N, P pξ “ kq “ ppkq where ppkq is given ř in (4.4.2), and Sk “ ki“1 ξi where ξ1 , ξ2 , . . . are independent copies of ξ. This section aims to describe the law of pξi : 1 ď i ď N pnqq conditioned on SN pnq “ mpnq. As discussed at the start of last section, this is a random allocation problem, with mpnq unlabelled balls and N pnq labelled boxes in total, viewing ξi as the number of balls in the i-th box. There are many established results for balls-inboxes models where the number of balls is proportional to the number of boxes; see the survey by Janson [41]. However, here we need to allow the balls-to-boxes ratio mpnq N pnq

to tend to infinity, so a variant of established work is needed. We accomplish

this in Propositions 4.5.1 and 4.5.2, extending the result of [41, Theorem 19.34]. rpnq,N pnq

These bounds can be applied to a uniform quadrangulation in Qn

, by using

Proposition 4.4.1. Given k P N, for any sequence px1 , . . . , xk q Ă Rk , write pxk,p1q , . . . , xk,pkq q as its decreasing ordered sequence, using a fixed rule to break ties. We write pxk,p1q , . . . , xk,pkq q as pxp1q , . . . , xpkq q when the context is clear. In particular, we often write ξp1q “ ξN pnq,p1q “ maxpξi : 1 ď i ď N pnqq. Let ν “ E rξs; clearly, ν ă 8. The exact value of ν is irrelevant for the current chapter, but it is nonetheless derived in Section 4.12 for completeness.

Proposition 4.5.1. Suppose that lim supnÑ8 ˆ P ξp1q

P ξp1q

ă 1. Then as n Ñ 8,

˙ ` ´10 ˘ mpnq ˇˇ S “ mpnq “ O n . ď ˇ N pnq pln nq2

Proposition 4.5.2. Suppose that lim supnÑ8 ˆ

νN pnq mpnq

νN pnq mpnq

ă 1. Then as n Ñ 8,

˙ ˇ ` ˘ mpnq 5 5{6 ˇ ´5{4 ą , ξ ą rpnq S “ mpnq “ O N pnq pln nq rpnq . ˇ p2q N pnq pln nq2

Before proving Propositions 4.5.1 and 4.5.2, we state an immediate application to quadrangulations, below. Recall the definitions of Qr,k n from (4.4.3) and Yi p¨q from (4.3.2).

107

4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

Corollary 4.5.3. Suppose that lim supnÑ8

νN pnq mpnq

108

rpnq,N pnq

ă 1. Then for Qn Pu Qn

as n Ñ 8, ˆ

mpnq P Yp1q pQn q ď pln nq2

(4.5.1)

˙ ` ˘ “ O n´10 ,

and ˆ

mpnq P Yp1q pQn q ą , Yp2q pQn q ą rpnq5{6 2 pln nq

˙ ` ˘ “ O N pnq pln nq5 rpnq´5{4 ;

it follows that (4.5.2)

` ˘ ` ˘ P Yp2q pQn q ą rpnq5{6 “ O N pnq pln nq5 rpnq´5{4 .

Proof. The first two equalities follow immediately from Propositions 4.4.1, rpnq,N pnq

4.5.1, and 4.5.2. For the last assertion, simply note that for Qn Pu Qn

,

` ˘ P Yp2q pQn q ą rpnq5{6 ˆ ˙ ˆ ˙ mpnq mpnq 5{6 , Yp2q pQn q ą rpnq ` P Yp1q pQn q ď ď P Yp1q pQn q ą pln nq2 pln nq2 ` ˘ ` ˘ “ O N pnq pln nq5 rpnq´5{4 ` O n´10 ` ˘ “ O N pnq pln nq5 rpnq´5{4 .



Corollary 4.5.3 gives size-bounds for the largest and second largest submaps rpnq,N pnq

pendant to the root block of a uniform quadrangulation in Qn

. In the next

section, we deduce a bound on the number of facial 2-cycles in the pre-root-block rpnq

of a uniform quadrangulation in Qn , which entails us to apply the bounds in Corollary 4.5.3 to the latter setting. Now we turn to establishing Propositions 4.5.1 and 4.5.2, starting with two lemmas related to sums of asymptotically stable distributions. For k, ` P N, write ξ`k “ ξ` 1rξ` ďks

,

4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

109

and S`k

` ÿ



ξik .

i“1

Lemma 4.5.4. For m P N and x ą 0, as k Ñ 8 we have ˘ ` k x νm ě x ď e´ k ` k p1`op1qq . P Sm Proof. By Chernorff inequality and by the fact that pξik : i P Nq are iid, for x ą 0 and s ą 0, (4.5.3)

´ ” k ı¯m ” kı ` k ˘ P Sm ě x ď e´sx E esSm “ e´sx E esξ1 .

Furthermore, k ” kı ÿ sξ1 E e “ P pξ ą kq ` P pξ “ tq ¨ est t“1

“ 1`

k ÿ

` ˘ P pξ “ tq ¨ est ´ 1

t“1

ď 1 ` sν `

k ÿ

` ˘ P pξ “ tq ¨ est ´ 1 ´ st .

t“1

By (4.4.2), there exists c ą 0, not depending on k, such that for all t ě 1, P pξ “ tq ď ct´5{2 . It follows from Taylor expansion that, when st ď 1 we have est ´1´st ď s2 t2 . So ”

sξ1k

ı

E e

ď 1 ` sν ` c

k ÿ

t´5{2 s2 t2 .

t“1

Now take s “ k1 , then k ÿ t“1

´5{2 2 2

t

st “

k ÿ t´1{2 t“1

k2

ˆ “O

k 1{2 k2

˙

ˆ ˙ 1 “o k

as k Ñ 8. Combining the bounds in the previous two displays, we have (4.5.4)

„ k ˆ ˙ ξ1 ν ν 1 E ek ď1` `o ď e k p1`op1qq . k k

With s “ k1 , the lemma follows immediately from (4.5.3) and (4.5.4).



4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

Recall that Sk “

řk

i“1 ξi

for k P N and that ν “ E rξs.

Lemma 4.5.5. Fix λ P p0, 1q. There exists δ “ δpλq ą 0 such that the following holds. For all sufficiently large integers N and m with λm ě νN “ E rSN s, we have P pSN “ mq ě

δN . m5{2

Proof. Fix large enough integers N and m with λm ě νN and N 2{3 ă

p1´λqm . 2

For each i “ 1, . . . , N , let " Ei “ |m ´ ξi ´ νN | ă N

2{3

* p1 ´ λqm , SN “ m . , max ξj ď j“1,...,i´1,i`1,...,N 2

Since νN ď λm, if the event Ei occurs, then ξi ą m ´ νN ´ N 2{3 ě p1 ´ λqm ´ N 2{3 ą

p1 ´ λqm ě max ξj . j“1,...,i´1,i`1,...,N 2

Therefore, the events E1 , . . . , EN are disjoint. It follows by symmetry and independence that P pSN “ mq ě N ¨ P pEN q ` ˘ ě N ¨ P |m ´ ξN ´ νN | ă N 2{3 , SN “ m ˙ ˆ p1 ´ λqm p1 ´ λqm 2 , ξN ´1 ą , SN “ m ´ N ¨ P ξN ą 2 2 “N

tνN ÿ `N 2{3 u

P pSN ´1 “ kq P pξN “ m ´ kq

k“rνN ´N 2{3 s

´N

2

m ÿ

ˆ P ξN ´1

`“r p1´λqm s 2

ěN

tνN ÿ `N 2{3 u

˙ p1 ´ λqm ą , SN ´1 “ m ´ ` P pξN “ `q 2

P pSN ´1 “ kq P pξ “ m ´ kq

k“rνN ´N 2{3 s

ˆ

p1 ´ λqm ´N ¨P ξ ą 2 2

˙ ¨

sup

P pξ “ `q .

r p1´λqm sď`ďm 2

So with ` ˘ c “ inf `5{2 ¨ P pξ “ `q : ` P N ą 0 ,

110

4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

for all k in the above sum, we have P pξ “ m ´ kq ě

c . m5{2

111

Similarly, with

` ˘ d “ sup `5{2 ¨ P pξ “ `q : ` P N ă 8 , ´ ¯ ´ ¯3{2 2d 2 we have P ξ ą p1´λqm we have P pξ “ `q ď ď , and for ` ě p1´λqm 2 3 p1´λqm 2 ´ ¯5{2 2 d ď d p1´λqm . Together with the preceding inequalities, this yields that `5{2 P pSN “ mq Nc ě 5{2 m

tνN ÿ `N 2{3 u

P pSN ´1 k“rνN ´N 2{3 s

2d “ kq ´ N ¨ 3 2

ˆ

2 p1 ´ λqm

˙3{2

ˆ ¨d

2 p1 ´ λqm

˙5{2 .

Since ξ is in the domain of attraction of a 32 -stable random variable, the fluctuation of SN ´1 around its mean is of order N 2{3 . By decreasing c if necessary, we may thus also assume that ` ˘ P |SN ´1 ´ νN | ă N 2{3 ě c , and obtain that c2 N 2d2 P pSN “ mq ě 5{2 ´ m 3

ˆ

2 1´λ

˙4

N2 c2 N ě ; m4 2m5{2

the last inequality holding since for any ε ą 0, we have

N2 m4

ă

εN m5{2

for large enough

N and for all m permitted by the lemma.

k Proof of Proposition 4.5.1. First, fix k P N, and recall that SN pnq “



řN pnq i“1

ξik

where ξik “ ξi 1rξi ďks . Considering which summand of SN pnq is largest leads to the inclusion of events tξp1q “ ku X tSN pnq “ mpnqu Ă

Nď pnq

k k tξi “ ku X tSN pnq ´ ξi “ mpnq ´ ku .

i“1

Write w “

ř8 `“1

wp`q. By symmetry and independence it follows that

` ˘ ` ˘ k P ξp1q “ k, SN pnq “ mpnq ď N pnq ¨ P ξN pnq “ k, SN pnq´1 “ mpnq ´ k ` ˘ ` k ˘ “ N pnq ¨ P ξN pnq “ k P SN “ mpnq ´ k pnq´1

4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

“ N pnq ¨ Since lim supnÑ8

νN pnq mpnq

˘ ` k 1 ` ψpkq “ mpnq ´ k . ¨ P S N pnq´1 k 5{2 w

ă 1, there exists λ P p0, 1q such that lim supnÑ8

νN pnq mpnq

ď λ.

By Lemma 4.5.5 there exists δ “ δpλq ą 0 such that for all n large enough we have ` ˘ δN pnq P SN pnq “ mpnq ě mpnq 5{2 , so Bayes formula now gives ´ P ξp1q

ˇ ¯ 1 ` ψpkq ˆ mpnq ˙5{2 ˘ ` k ˇ “ k ˇ SN pnq “ mpnq ď ¨ P SN pnq´1 “ mpnq ´ k . δw k

Note that if SN pnq “ mpnq then ξp1q ě mpnq{N pnq. Therefore, with N pnq ă pln nq2 ˇ ´ ¯ mpnq ˇ we have P ξp1q ď pln nq2 ˇ SN pnq “ mpnq “ 0. It thus remains to consider the case N pnq ě pln nq2 : ˆ P ξp1q t

˙ mpnq ˇˇ ď ˇ SN pnq “ mpnq pln nq2

mpnq u pln nq2

1 ` ψpkq δw

ÿ ď k“t mpnq u N pnq

N pnq5{2 ď δw

ˆ

mpnq k

˙5{2 ` k ˘ ¨ P SN pnq´1 “ mpnq ´ k ˆ

sup t mpnq uďkďt N pnq

k SN pnq´1

p1 ` ψpkqq ¨ P mpnq u pln nq2

ˆ

˙˙

ˆ

˙˙

1 ě mpnq 1 ´ pln nq2

(4.5.5) N pnq5{2 “ δw

ˆ sup t mpnq uďkďt N pnq

t

mpnq u pln nq2

p1 ` ψpkqq ¨ P SN pnq´1

mpnq u pln nq2

Since mpnq “ np1`op1qq, we have

mpnq pln nq2

1 ě mpnq 1 ´ pln nq2

Ñ 8 as n Ñ 8. Therefore, by Lemma 4.5.4,

as n Ñ 8, ˙˙ 1 P SN pnq´1 ě mpnq 1 ´ pln nq2 ˜ ¸ ˆ ˙ mpnq 1 νN pnq ď exp ´ mpnq 1´ ` mpnq p1 ` op1qq . pln nq2 t pln nq2 u t pln nq2 u ˆ

Since lim supnÑ8 ´

t

mpnq u pln nq2

νN pnq mpnq

mpnq mpnq t pln u nq2

ˆ

ă 1, there is ε ą 0 such that for large enough n,

ˆ

1 1´ pln nq2

˙ `

.

νN pnq mpnq t pln u nq2

p1 ` op1qq ă ´ εpln nq2 .

112

4.5. RANDOM ALLOCATION WITH VARYING BALLS-TO-BOXES RATIO

113

Hence, as n Ñ 8, ˆ

t

mpnq u pln nq2

P SN pnq´1

ˆ

1 ě mpnq 1 ´ pln nq2

˙˙ 2

ă e´εpln nq .

This combined with (4.5.5) yields ˆ P ξp1q

˙ ˙ ˆ ` ˘ mpnq ˇˇ N pnq5{2 ď “ O n´10 . ˇ SN pnq “ mpnq “ O 2 ε ln n pln nq n



Proof of Proposition 4.5.2. By symmetry and independence, ˆ

˙ ˇ mpnq 5{6 ˇ P ξp1q ą , ξp2q ą rpnq ˇ SN pnq “ mpnq pln nq2 ˙ ˆ ˇ mpnq 5{6 ˇ 2 , ξN pnq´1 ą rpnq ˇ SN pnq “ mpnq ď N pnq ¨ P ξN pnq ą pln nq2 ` ˘ ` ˘ mpnq ÿ P ξN pnq “ i P ξN pnq´1 ą rpnq5{6 , SN pnq´1 “ mpnq ´ i 2 ` ˘ “ N pnq P S “ mpnq N pnq mpnq i“t

`1u pln nq2

˜ˆ 2

“ N pnq ¨ O ˜ˆ 2

ď N pnq ¨ O

mpnq pln nq2

˙´5{2 ¸

mpnq pln nq2

˙´5{2 ¸

mpnq ÿ

¨ i“t

mpnq `1u pln nq2

` ˘ P ξN pnq´1 ą rpnq5{6 , SN pnq´1 “ mpnq ´ i ` ˘ P SN pnq “ mpnq

` ˘ P ξN pnq´1 ą rpnq5{6 ` ˘ ; ¨ P SN pnq “ mpnq

in the second equality we use the fact that for i ě

mpnq , pln nq2

ˆ´ P pξ “ iq “ O

mpnq pln nq2

¯´5{2 ˙ .

Furthermore, since ξ is in the domain of attraction of a 23 -stable random variable, we ´ ¯ ` ˘ ` ˘ ´ 32 ¨ 56 5{6 have P ξ ą rpnq “ O rpnq “ O rpnq´5{4 . Together with Lemma 4.5.5, ˆ

˙ ˇ mpnq 5{6 ˇ , ξp2q ą rpnq ˇ SN pnq “ mpnq P ξp1q ą pln nq2 ˜ ¸ ˆ ˙´5{2 mpnq 1 ˘ “ O N pnq2 rpnq´5{4 ¨ ` 2 pln nq P SN pnq “ mpnq ¸ ˜ ˆ ˙´5{2 mpnq “ O N pnq2 rpnq´5{4 N pnq´1 mpnq5{2 pln nq2 ` ˘ “ O N pnq pln nq5 rpnq´5{4 .



4.6. THE NUMBER OF FACIAL 2-CYCLES IN THE PRE-ROOT-BLOCK

4.6. The Number of Facial 2-Cycles in the Pre-Root-Block This section shows that for Qn Pu Qrn with appropriate r, we have |FpQn q| ă 3r with high probability as n Ñ 8, recalling that |FpQn q| is the number of facial 2-cycles in the pre-root-block of Qn . Together with the assumptions that rpnq ą pln nq25 and rpnq “ opnq, this verifies that the conditions in Corollary 4.5.3 hold with high probability, paving the way to proving condensation phenomena for Qn in Section 4.7. Recall that ν “ E rξs. Proposition 4.6.1. Fix λ P p0, 1q. There exists c “ cpλq ą 0 such that the following holds. For all sufficiently large integers r and n with λ ¨ pn ´ rq ě 2νr, given Qn Pu Qrn , we have ˆ ˙r 4 P p|FpQn q| ě 3rq ď c ¨ ¨ n5{2 . 9 We first derive several lemmas before presenting the proof for Proposition 4.6.1, shown in the end of this section. Recall the definition of Qr,k n from (4.4.3), and recall that Rr denotes the set of rooted 2-connected quadrangulations with r vertices. Lemma 4.6.2. For n, r, k P N with r ă n and k ď n ´ r, we have ˆ ˙ ˇ r,k ˇ 2r ´ 4 ` k ˇQn ˇ “ |Rr | ¨ ¨ 2k k y

ÿ 1 `...`yk “n´r

k ź

|Qyi `1 | .

i“1

Lemma 4.6.2 follows from Lemma 4.4.2 by summing over sequences of y “ py1 , . . . , yk q P Bn´r,k with y1 ` . . . ` yk “ n ´ r. Next, let M pzq be the generating function of rooted quadrangulations with z marking the number of faces (or the number of vertices minus two). That is, M pzq “

8 ÿ

|Q``2 | ¨ z ` .

`“1

|Q``2 | “

`1˘

1 3

by [16, Proposition 4]. Furthermore, by (4.4.1) we have

12` p1`ψp``1qq ?2 π p``1q5{2

for ` P N, and we take |Q2 | “ 1 since we view a single edge

Note that M

12



114

4.6. THE NUMBER OF FACIAL 2-CYCLES IN THE PRE-ROOT-BLOCK

115

as a quadrangulation. So ˆ ˙ 8 8 ÿ 4 2 1 ` ψp`q ÿ 1 1 ? “ . “ |Q``2 | ¨ ` “ 1 ` M 5{2 π ` 12 12 3 `“1 `“0 Thus,

1`ψp`q `“1 `5{2

ř8



? 2 π . 3

Recalling the distribution of ξ from (4.4.2), we then have,

for i P N, 1`ψpiq

P pξ “ iq “ ř8 i

(4.6.1)

5{2

1`ψp`q `5{2

`“1

Recall that S` “

ř`

i“1 ξi

3 1 ` ψpiq . “ ? 2 π i5{2

for ` P N, where ξ1 , ξ2 , . . . are independent copies of ξ.

Corollary 4.6.3. Fix n, r, k P N with r ă n and k ď n ´ r. Then ˙ ˆ ˙k 2r ´ 4 ` k 2 12n´r . “ |Rr | ¨ ¨ P pSk “ n ´ rq ¨ 9 k ˆ

|Qr,k n |

Proof. It follows from (4.4.1) that k ź

ÿ

ˆ |Qyi `1 | “

y1 `...`yk “n´r i“1

2 ? π

˙k 12n´r´k

ÿ

k ź 1 ` ψpyi q 5{2

y1 `...`yk “n´r i“1

.

yi

Combined with (4.6.1), we have k ź

ÿ

ˆ |Qyi `1 | “

y1 `...`yk “n´r i“1

2 ? π

˙k n´r´k

12

ˆ ? ˙k 2 π ¨ P pSk “ n ´ rq . 3

The result then follows from Lemma 4.6.2.



Lemma 4.6.4. Fix n, r, k P N with r ă n and 2r ´ 4 ď k ď n ´ r. Then ˇ r,k ˇ ˇ r,2r´4 ˇ ˇQn ˇ ď ˇQn ˇ¨

ˆ ˙k´2r`4 P pSk “ n ´ rq 4 ¨ . 9 P pS2r´4 “ n ´ rq

Proof. Note that for all a P N, ˆ

Since

a`k k

ˆ ˙ ˙ a`k`1 pa ` k ` 1q! a`k`1 a`k “ “ . a!pk ` 1q! k`1 k`1 k

decreases in k, it follows from Corollary 4.6.3 that for k ě 2r ´ 4 we have ˆ

ˇ r,k ˇ ˇ ˇ ˇQn ˇ ď ˇQr,2r´4 ˇ¨ n

2r ´ 4 ` 2r ´ 4 2 ¨ 2r ´ 4 9

˙k´2r`4 ¨

P pSk “ n ´ rq P pS2r´4 “ n ´ rq

4.7. CONDENSATION IN UNIFORM QUADRANGULATION CONDITIONED ON ROOT BLOCK SIZE 116 ˇ ˇ ˇ¨ ď ˇQr,2r´4 n

ˆ ˙k´2r`4 4 P pSk “ n ´ rq ¨ . 9 P pS2r´4 “ n ´ rq



Proof of Proposition 4.6.1. Fix large enough integers n and r with λ¨pn´rq ě 2νr. It follows from Lemma 4.5.5 that there exists δ “ δpλq ą 0 such that P pS2r´4 “ n ´ rq ě

δ ¨ p2r ´ 4q . n5{2

Since probabilities are at most 1, the bound in Lemma 4.6.4 gives that, for 2r ´ 4 ď k ď n ´ r, ˆ ˙k´2r`4 n5{2 4 9 δ ¨ p2r ´ 4q ˆ ˙k´2r`4` 52 log 4 n ˇ r,2r´4 ˇ 4 1 9 ˇ ˇ ď Qn ¨ . 9 δ

ˇ r,k ˇ ˇ ˇ ˇQn ˇ ď ˇQnr,2r´4 ˇ ¨

In particular, for ` P N, ˇ ˇ ˆ ˙` ˇ r,2r´4´ 52 log 4 n`` ˇ ˇ r,2r´4 ˇ 9 ˇQn ˇ ď ˇQ n ˇ¨ 4 1 . ˇ ˇ 9 δ So for Qn Pu Qrn , P p|FpQn q| ě 3rq “

|Qr,k n | řn´r r,` `“0 |Qn | 3rďkďn´r ÿ

ˇ r,k ˇ ˇQ n ˇ

ÿ ď 2r´4´ 52

ď

9 5δ



9 5δ

log 4 n`r` 52 9

log 4 nďkďn´r

ˆ ˙r` 52 log 4 n 4 9 9 ˆ ˙r 4 n5{2 . 9

ˇ r,2r´4 ˇ ˇQ n ˇ

9



4.7. Condensation in Uniform Quadrangulation Conditioned on Root Block Size rpnq

In this section, we show that for Qn Pu Qn , with high probability, there is condensation in Qn (see [3] for an overview of condensation in random maps), and

4.7. CONDENSATION IN UNIFORM QUADRANGULATION CONDITIONED ON ROOT BLOCK SIZE 117 the root block RpQn q does not separate two large submaps; precise statements appear in Corollaries 4.7.1 and 4.7.2 respectively. Recall from (4.3.2) that pYi pQn q ` 1 : 1 ď i ď |FpQn q|q are the sizes of submaps pendant to RpQn q, where |FpQn q| is the number of facial 2-cycles in the pre-rootblock of Qn . We write Yp1q pQn q, Yp2q pQn q, . . . , Yp|F pQn q|q pQn q in its decreasing order, using a fixed rule to break ties.

rpnq

Corollary 4.7.1. For Qn Pu Qn , as n Ñ 8, ˆ

mpnq P Yp1q pQn q ď pln nq2

˙ “ op1q .

rpnq

Corollary 4.7.2. For Qn Pu Qn , as n Ñ 8, ` ˘ P Yp2q pQn q ą rpnq5{6 “ op1q . Corollaries 4.7.1 and 4.7.2 are immediate consequences of Proposition 4.6.1, (4.5.1), and (4.5.2). Their proofs are similar, so we only present the latter one.

rpnq

Proof of Corollary 4.7.2. Fix Qn Pu Qn . We have ` ˘ P Yp2q pQn q ą rpnq5{6 ˇ ´ ¯ ˇ ď P Yp2q pQn q ą rpnq5{6 ˇ |FpQn q| ă 3rpnq ` P p|FpQn q| ě 3rpnqq . Recalling that rpnq ą pln nq25 for all n and rpnq “ opnq as n Ñ 8, it follows from Proposition 4.6.1 that P p|FpQn q| ě 3rpnqq “ op1q as n Ñ 8. Furthermore,

ˇ ´ ¯ ˇ P Yp2q pQn q ą rpnq5{6 ˇ |FpQn q| ă 3rpnq ˇ ¯ ´ ˇ ď sup P Yp2q pQn q ą rpnq5{6 ˇ Qn P Qrpnq,k . n kă3rpnq

4.7. CONDENSATION IN UNIFORM QUADRANGULATION CONDITIONED ON ROOT BLOCK SIZE 118 It follows from (4.5.2) that, for N pnq ă 3rpnq, ˇ ´ ¯ ` ˘ ˇ P Yp2q pQn q ą rpnq5{6 ˇ Qn P Qnrpnq,N pnq “ O N pnqpln nq5 rpnq´5{4 ` ˘ “ O rpnq´1{4 pln nq5 . Thus, ˇ ¯ ´ ` ˘ 5{6 ˇ rpnq,k “ O rpnq´1{4 pln nq5 “ op1q ; sup P Yp2q pQn q ą rpnq ˇ Qn P Qn

kă3rpnq

the last equality follows from the assumption that rpnq ą pln nq25 , completing the proof.

 rpnq

Next, for Qn Pu Qn , we derive a tail bound for the maximal diameter of nonlargest submap pendant to the root block. The derivation relies on [28, Proposition 4], stated as follows. Proposition 4.7.3. (Chassaing & Schaeffer [28]). Fix n P N and let Qn Pu Qn . There exist positive constants x0 , c1 , and c2 such that for all x ą x0 , ` ˘ P diam pQn q ą xn1{4 ď c1 e´c2 x . Combining Corollary 4.7.2 and Proposition 4.7.3, we obtain the following corollary easily. Given Q P Q, recall from (4.3.1) that pPi pQq : 1 ď i ď |FpQq|q is the sequence of submaps pendant to RpQq. We write them in the decreasing order of size as pPpiq pQq : 1 ď i ď |FpQq|q, using a fixed rule to break ties. rpnq

Corollary 4.7.4. Let Qn Pu Qn . Then for all x ą 0, as n Ñ 8, ˆ P

˙ max 2ďiď|F pQn q|

1{4

diampPpiq pQn qq ě x ¨ rpnq

“ op1q .

Proof. For |FpQn q| ă i ď n ´ rpnq, write Ppiq pQn q “ H. Let k P Ně0 with k ď n ´ r ` 1. For any 1 ď i ď n ´ rpnq, given that |vpPpiq pQn qq| “ k, Ppiq pQn q is uniformly distributed over Qk (denoting Q0 “ H). By Proposition 4.7.3, there exist

4.7. CONDENSATION IN UNIFORM QUADRANGULATION CONDITIONED ON ROOT BLOCK SIZE 119 positive constants x0 , c1 , and c2 such that for all x ě x0 , and for all 1 ď i ď n´rpnq, ˇ ´ ¯ ˇ P diampPpiq pQn qq ě xk 1{4 ˇ |vpPpiq pQn qq| “ k ď c1 e´c2 x .

(4.7.1)

Fix x ą 0. We have ˙

ˆ P

max 2ďiďn´rpnq

diampPpiq pQn qq ě x ¨ rpnq

1{4

˙

ˆ ďP

max 2ďiďn´rpnq

diampPpiq pQn qq ě x ¨ rpnq

1{4

5{6

, Yp2q pQn q ď rpnq

` ˘ ` P Yp2q pQn q ą rpnq5{6 .

By Corollary 4.7.2, ` ˘ P Yp2q pQn q ą rpnq5{6 “ op1q . Next, using a union bound, ˆ

˙ max

P

2ďiďn´rpnq

n´rpnq ÿ

ď n´rpnq ÿ i“2

5{6

, Yp2q pQn q ď rpnq

` ` ˘ ˘ P diam Ppiq pQn q ě x ¨ rpnq1{4 , |vpPpiq pQn qq| “ k

sup 1ďkďtrpnq5{6 u

i“2

(4.7.2) ď

diampPpiq pQn qq ě x ¨ rpnq

1{4

sup 1ďkďtrpnq5{6 u

ˇ ´ ¯ ` ˘ 1{4 ˇ P diam Ppiq pQn q ě x ¨ rpnq ˇ |vpPpiq pQn qq| “ k .

By (4.7.1), for all 1 ď k ď rpnq5{6 and 1 ď i ď n ´ rpnq, ˇ ´ ¯ ` ˘ ` ˘ 1{4 ˇ P diam Ppiq pQn q ě x ¨ rpnq ˇ |vpPpiq pQn qq| “ k ď c1 ¨ exp ´c2 x ¨ rpnq1{4 k ´1{4 ` ˘ ď c1 ¨ exp ´c2 x ¨ rpnq1{24 . Thus, it follows from (4.7.2) that ˆ P

˙ max 2ďiďn´rpnq

1{4

diampPpiq pQn qq ě x ¨ rpnq

` ˘ ď nc1 ¨ exp ´c2 x ¨ rpnq1{24 ` ˘ “ c1 ¨ exp ln n ´ c2 x ¨ rpnq1{24 .

5{6

, Yp2q pQn q ď rpnq

4.8. UNIFORMLY ASYMPTOTICALLY NEGLIGIBLE ATTACHMENTS

120

By the assumption that rpnq ą pln nq25 , we have ln n ´ c2 x ¨ rpnq1{24 Ñ ´8

(4.7.3)

as n Ñ 8, completing the proof.



4.8. Uniformly Asymptotically Negligible Attachments Fix Qn Pu Qn . Recall from Section 4.1.1 that R` pQn q “ Qn ´ v pΛpQn qq ztρQn u. In what follows, Propositions 4.8.1 and 4.8.2 show that R` pQn q and RpQn q have the same scaling limit, when respectively endowed with the measures in Theorem 4.1.2 and 4.1.1. With the measure of Theorem 4.1.2, no mass is assigned to the components of R` pQn q ´ vpRpQn qq, so the convergence in Proposition 4.8.1 is easier to establish than that in Proposition 4.8.2. Given a graph G, we often write dG for the graph distance on any induced subgraph of G, and for the intrinsic metric in its approximating boundedly compact length space. ´ Recall that kn “

40¨rpnq 21

¯1{4 for n P N. rpnq

Proposition 4.8.1. For Qn Pu Qn , we have ˆ (4.8.1)

1 1 ¨ dQn , ρQn , ¨ νRpQn q R pQn q, kn rpnq `

˙ Ñ m8

as n Ñ 8 in distribution for the pointed Gromov-Hausdorff-Prokhorov topology. rpnq

Proof. For n P N, let Qn Pu Qn . We write ˆ pn “ R ˆ p` “ R n

1 1 RpQn q, ¨ dQn , ρQn , ¨ νRpQn q kn rpnq

˙

1 1 R pQn q, ¨ dQn , ρQn , ¨ νRpQn q kn rpnq `

, ˙ .

d pn Ñ As discussed in Section 4.1.3, R m8 for the pointed GHP topology, so it suffices ´ ¯ p p n, R p` Ñ to show that d‹GHP R 0. n

4.8. UNIFORMLY ASYMPTOTICALLY NEGLIGIBLE ATTACHMENTS

p n and R p` Write On for the set of components of R` pQn q ´ vpRpQn qq. Since R n are equipped with the same measure, it follows from the definition of d‹GHP that ´ ¯ p n, R p ` ď 1 ¨ max diampGq . d‹GHP R n kn GPOn By Corollary 4.7.4, maxGPOn diampGq “ oprpnq1{4 q with 1 ´ op1q probability. Since ´ ¯ p p n, R p` Ñ kn “ Θprpnq1{4 q, it follows that d‹GHP R 0, completing the proof.  n rpnq

Proposition 4.8.2. For Qn Pu Qn , we have ˆ (4.8.2)

1 1 R pQn q, ¨ dQn , ρQn , ¨ νR` pQn q ` kn |vpR pQn qq| `

˙ Ñ m8

as n Ñ 8 in distribution for the pointed Gromov-Hausdorff-Prokhorov topology. We devote the rest of this section to proving Proposition 4.8.2. We proceed in rpnq

the following two steps to show that given Qn Pu Qn , with high probability there are no w P vpRpQn qq to which an overly large mass of pendant submaps attach. First, in Lemma 4.8.3 we prove a tail bound for the maximum degree in a uniform rooted 2-connected quadrangulation. Secondly, we prove that with high probability no edge of RpQn q is subdivided many times in Qn , as shown in Lemma 4.8.4. Given a graph G, for u P vpGq, write degG puq for the degree of u in G. Recall that Rr denotes the set of rooted 2-connected quadrangulations with r vertices. Lemma 4.8.3. Fix x P N. For any ε ą 1{2 there exists B ą 0 such that for all r P N, given Rr Pu Rr , ˆ P

˙ max degRr puq “ x

uPvpRr q

ď Bεx r5{3 .

Proof. This straightforward proof is a slight modification of the proof of [5, Lemma 8.2]. First, for any ε ą 1{2 there exists c “ cpεq ą 0 such that for all n P N, given Qn Pu Qn , ˆ (4.8.3)

P

˙ max degQn pvq “ x

vPvpQn q

ď cεx n ;

121

4.8. UNIFORMLY ASYMPTOTICALLY NEGLIGIBLE ATTACHMENTS

122

see [18, Theorem 2 (i)]. Now, fix r P N and write n “ t15r{7u. Let Rr Pu Rr and Qn Pu Qn . Next, let R1 pQn q be the largest block of Qn , rooted at its ăQn minimal edge; if there are multiple blocks of size |vpR1 pQn qq|, then among these blocks we choose R1 pQn q to be the one whose root edge is ăQn -minimal. Given that |vpR1 pQn qq| “ r, R1 pQn q has the same law as Rr . So ˆ P

˙ max degRr puq “ x

uPvpRr q

(4.8.4)

ˆ

˙ ˇ ˇ 1 “P max degR1 pQn q pvq “ x ˇ |vpR pQn qq| “ r vPvpR1 pQn qq ˆ ˙ P max degR1 pQn q pvq “ x vPvpR1 pQn qq ď P p|vpR1 pQn qq| “ rq ˆ ˙ P max degQn pvq “ x vPvpQn q . ď P p|vpR1 pQn qq| “ rq

Note that n “ t15r{7u. By [5, Proposition 4.3], there thus exists c1 ą 0 such that P p|vpR1 pQn qq| “ rq ě c1 r´2{3 . Together with (4.8.3) and (4.8.4), it follows that for all n P N, ˙

ˆ P rpnq

Fix Qn P Qn

max degRr puq “ x

uPvpRr q

ď

c x 2{3 15c ε r n ď 1 εx r5{3 . 1 c 7c



for now. List the edges of RpQn q as e1 , . . . , e2rpnq´4 in ăQn -order.

Create an extra copy e0 of e1 as in the decomposition described in the proof of Lemma 4.4.2. For i “ 0, . . . , 2rpnq ´ 4, write `i pQn q for the number of copies of ei in Qn minus one, that is, `i pQn q is the number of facial 2-cycles resulting from the split of ei . rpnq

Lemma 4.8.4. For Qn Pu Qn , as n Ñ 8 we have ˆ P

max 0ďiď2rpnq´4

˙ ` ˘ `i pQn q ą 5 ln rpnq “ O rpnq´1 . rpnq

Proof. For n P N let Qn Pu Qn . Note that

ř2rpnq´4 i“0

`i pQn q “ |FpQn q|. It

follows that the vector p`0 pQn q, . . . , `2rpnq´4 q is distributed as a uniformly random weak composition of |FpQn q| into 2rpnq´3 parts. (Recall that in a weak composition,

4.8. UNIFORMLY ASYMPTOTICALLY NEGLIGIBLE ATTACHMENTS

123

empty parts are allowed.) In particular, `0 pQn q is distributed as the size of the first part in such a composition. Using that the number of weak compositions of a into b ´ ¯ ` ˘ b´1 is a`b´1 , and noting that 1 ´ is increasing in j, it follows that for integers b´1 j`b´1 j ă k, ´ P `0 pQn q ą j

ˆ ˇ ¯ ˇ ˇ |FpQn q| “ k ď 1 ´

2rpnq ´ 4 k ` 2rpnq ´ 4

˙j .

Moreover, it follows from Proposition 4.6.1 that ´ ¯ rpnq 5{2 P p|FpQn q| ě 3rpnqq “ O p4{9q ¨n as n Ñ 8. Recalling that rpnq ą pln nq25 , it follows that P p`0 pQn q ą 5 ln rpnqq ˇ ´ ¯ ÿ ˇ P `0 pQn q ą 5 ln rpnq ˇ |FpQn q| “ k P p|FpQn q| “ kq ` P p|FpQn q| ě 3rpnqq ď kă3rpnq

˙5 ln rpnq ´ ¯ 2rpnq ´ 4 ď 1´ p1 ´ op1qq ` O p4{9qrpnq ¨ n5{2 3rpnq ` 2rpnq ´ 4 ´ ¯ ď e´2 ln rpnq p1 ´ op1qq ` O p4{9qrpnq ¨ n5{2 ˆ

` ˘ “ O rpnq´2 . Finally, by a union bound, ˆ P

max 0ďiď2rpnq´4

˙ `i pQn q ą 5 ln rpnq ď 2rpnq ¨ P p`0 pQn q ą 5 ln rpnqq ` ˘ “ O rpnq´1 .



The next two facts provide deterministic bounds on the pointed GHP distance. Versions of these facts which apply to the non-pointed GHP distance appear in [5, Facts 7.3 and 7.4], and we omit their proofs.

Fact 4.8.5. Fix a pointed measured metric space V “ pV, d, o, µq and let W Ă V with o P W . Let µW be a non-negative finite Borel measure on pW, dq, and write

4.8. UNIFORMLY ASYMPTOTICALLY NEGLIGIBLE ATTACHMENTS

124

W “ pW, d, o, µW q. Then d‹GHP pV, Wq ď max tdH pV, Wq, dP pµ, µW qu . Fact 4.8.6. Fix a pointed measured metric space V “ pV, d, o, νq. Let W Ă V be finite with o P W so that there exists ε ą 0 with V “ tu P V : dpu, W q ď εu. Let Ť tPw : w P W u be such that wPW Pw “ V , that νpPw X Pw1 q “ 0 for w ‰ w1 , and that Pw Ă tu P V : dpu, wq ď εu for all w P W . Define a measure µ on W by setting µpwq “ νpPw q for any w P W , and let W “ pW, d, o, µq. Then d‹GHP pV, Wq ď ε . The final ingredient for proving Proposition 4.8.2 is an asymptotic bound on the L´evy-Prokhorov distance between the uniform measure on the vertices of a graph and a certain exchangeable perturbation of this measure. This is a reprise of [5, Lemma 6.3, Corollaries 7.1 and 7.2]. We start by introducing notation. In the sequel, for n “ pn1 , . . . , nn q P Rn and p ą 0, write |n|p “ p

řn i“1

1{p

npi q

.

Now, fix n P N, and let n “ pn1 , . . . , nn q be a vector of non-negative real numbers with |n|1 ą 0. Fix a rooted graph G P Gn , and list the vertices as v1 , . . . , vn in the ăG -order. Then define a measure on vpGq by setting, for V Ă vpGq, µnG pV q “

ÿ

ni .

ti:vi PV u

In words, we view ni as the total mass of pendant submaps attached to vi , and µnG as the measure assigning each vertex vi a mass of ni . ř Recall from Section 4.1.1 that νG “ vPvpGq δv is the counting measure. (Notice the different notations from previous chapter, where the measures were defined with renormalization.)

Lemma 4.8.7. For r P N, let n “ pn1 , . . . , nr q be an exchangeable random vector of non-negative real numbers, and let Rr Pu Rr . If |n|1 Ñ 8 and |n|2 {|n|1 Ñ 0 as

4.8. UNIFORMLY ASYMPTOTICALLY NEGLIGIBLE ATTACHMENTS

125

r Ñ 8, then ˆ dP

1 1 ¨ µnRr , ¨ νRr |n|1 r

˙ ` ˘ “ o r1{4

with 1 ´ op1q probability, where dP is the L´evy-Prokhorov distance on Rr .

Proof. Fix r P N, and write R “ Rr , for readability. List the vertices of R as v1 , . . . , vr in the ăR -order. It suffices to show that, for any V Ă vpRq and for any t ą 0, (4.8.5)

ˇ ˆˇ ˙ ˙ ˆ ˇ 1 2t2 2t ˇˇ ˇ 1 ˇ n P ˇ ¨ µ pV q ´ ¨ νR pV qˇ ą |n|2 ď 2 ¨ exp ´ 2 . |n|1 R r |n|1 ˇ |n|2

Assuming that (4.8.5) holds, Lemma 4.8.7 follows in a similar way as [5, Corollary 7.2] follows from [5, Lemma 6.3 and Corollary 7.1], and we refer the reader to that work for greater details. Now we turn to proving (4.8.5). Note that νR pV q “ |V |, and that fi » ˇ ÿ ˇ |V | . E– ni ˇˇ |n|1 fl “ |n|1 ¨ r ti:v PV u i

Then by a Hoeffding-type bound (see [54, Theorem 2.5]) we have ˆˇ ˇ 1 ˇ 1 ˇ n P ˇ ¨ µR pV q ´ ¨ νR pV qˇ ą |n|1 r ¨ ˇ ÿ |V | ˇˇ ˇ ni ´ |n|1 ¨ “ P ˝ˇ ˇ ą 2t r ti:vi PV u ˆ ˙ 2t2 ď 2 ¨ exp ´ 2 . |n|2

ˇ ˙ 2t ˇˇ |n|2 |n|1 ˇ ˛ ˇ ˇ ˇ |n|2 ‚ ˇ

This establishes (4.8.5) and thus concludes the proof.



rpnq

Proof of Proposition 4.8.2. Fix Qn Pu Qn . We write dn for the distance on any induced submap of Qn , let Rn` “ R` pQn q and Rn “ RpQn q. We root Rn at the ăQn -minimal edge, and write the resulting rooted map as Rn . Then list the vertices of Rn as v1 , . . . , vrpnq in the ăRn -order, noting that |vpRn q| “ rpnq.

4.8. UNIFORMLY ASYMPTOTICALLY NEGLIGIBLE ATTACHMENTS

126

Let On be the set of components of Rn` ´ vpRn q. For each v P vpRn q, let ď

Cv “

tvpGq : G P On , dn pG, vq “ 1u

ď

tvu .

Now, let n “ p|vpCvi q| : 1 ď i ď rpnqq. Note that µnRn pvq “ νRn` pCv q for v P vpRn q. Then let ˆ Rn “

1 1 Rn , ¨ dn , ρQn , ¨ µnRn ` kn |vpRn q|

ˆ p` “ R n

Rn` ,

˙ ,

1 1 ¨ dn , ρQn , ¨ν ` kn |vpRn` q| Rn

˙ .

It follows from Fact 4.8.6 that ´ ¯ p ` ď 1 ¨ max diampGq . d‹GHP Rn , R n kn GPOn By Corollary 4.7.4 and by the fact that kn “ Θprpnq1{4 q,

1 kn

p

¨ maxGPOn diampGq Ñ 0

as n Ñ 8. Hence, ´ ¯ p p` Ñ d‹GHP Rn , R 0. n

(4.8.6)

Furthermore, we claim that (4.8.7)

max |Cv | “ oprpnqq

vPvpRn q

with 1 ´ op1q probability; this claim is proven in the end of this proof. Note that ř |n|1 “ vPvpRn q |Cv | ą rpnq, and that rpnq Ñ 8 with n. Then (4.8.7) leads to ´ř |n|2 “ |n|1

vPvpRn q

|Cv |

ř vPvpRn q

2

¯1{2

|Cv |

˜ ď

maxvPvpRn q |Cv | ř vPvpRn q |Cv |

¸1{2 “ op1q

with 1 ´ op1q probability as n Ñ 8. This verifies the assumptions of Lemma 4.8.7. It thus follows from Lemma 4.8.7 that ˆ dP

1 1 ¨ µnRn , ¨ νRn |n|1 rpnq

˙ ` ˘ “ o rpnq1{4 “ o pkn q ,

with 1 ´ op1q probability, where dP denotes the L´evy-Prokhorov distance on Rn .

4.9. PROOFS OF THE MAIN THEOREMS

´ p n “ Rn , Next, let R

1 kn

¨ dn , ρ Q n ,

1 rpnq

127

¯ p n have the ¨ νRn . Note that Rn and R

same metric structure but with different measures, and that |vpRn` q| “ |n|1 . Then it follows from Fact 4.8.5 that d‹GHP

(4.8.8)

ˆ ˙ ´ ¯ 1 1 1 p n pn ď Rn , R ¨ dP ¨ µRn , ¨ νR n Ñ 0 . kn |n|1 rpnq

d pn Ñ As noted in Section 4.1.3, R m8 for the pointed GHP topology. Combined d p` Ñ with (4.8.6) and (4.8.8), we thus have R m8 for the pointed GHP topology, n

establishing (4.8.2). It remains to prove (4.8.7). Observe that max |Cv | ď

vPvpRn q

max 0ďiď2rpnq´4

`i pQn q ¨ max degRn pvq ¨ Yp2q pQn q ` 1 . vPvpRn q

By Corollary 4.7.2, Yp2q pQn q ď rpnq5{6 with 1 ´ op1q probability. Moreover, it follows from Lemma 4.8.3 that ˆ P

˙ ˘ ` max degRn pvq ě 3 ln rpnq “ O e´3 ln rpnq ¨ rpnq5{3 “ Oprpnq´1 q .

vPvpRn q

Finally, by Lemma 4.8.4, max0ďiď2rpnq´4 `i pQn q ď 5 ln rpnq with 1 ´ op1q probability. This combined with the bounds in the previous displays yields ` ˘ max |Cv | “ O ln rpnq ¨ ln rpnq ¨ rpnq5{6 “ oprpnqq

vPvpRn q

with 1 ´ op1q probability, establishing (4.8.7) and thus completing the proof.



4.9. Proofs of the Main Theorems Given pointed measured metric spaces X “ pX, d, x, µq and Y “ pY, d1 , y, µ1 q, let Z “ pXztxuq Y Y , and define a distance δ on Z by setting, for p, q P Z, $ ’ ’ dpp, qq ’ ’ ’ & δpp, qq “ d1 pp, qq ’ ’ ’ ’ ’ %dpp, xq ` d1 py, qq

if p, q P X if p, q P Y if p P X, q P Y

.

4.9. PROOFS OF THE MAIN THEOREMS

128

Then define a measure ν on the Borel set of pZ, δq by setting νpV q “ µpV XXztxuq` µ1 pV X Y q. Finally, let ZpX, Yq “ pZ, δ, y, νq. In words, ZpX, Yq is the pointed measured metric space obtained from X and Y by identifying the distinguished points of X and Y. Recall that given a pointed measured metric space V “ pV, d, o, νq, we let Br “ ´ ˇ ¯ Br pVq “ tw P V : dpw, oq ď ru, and write Br pVq “ Br , d, o, ν ˇBr . Lemma 4.9.1. Given pointed measured metric spaces pXn : 1 ď n ď 8q and pYn : 1 ď n ď 8q, if dLGHP pXn , X8 q Ñ 0 and dLGHP pYn , Y8 q Ñ 0, then dLGHP pZpXn , Yn q, ZpX8 , Y8 qq Ñ 0 as n Ñ 8. Proof. Write Zn “ ZpXn , Yn q and Z8 “ ZpX8 , Y8 q. Let r ě 0. By construction, for n P N, d‹GHP pBr pZn q, Br pZ8 qq ď d‹GHP pBr pXn q, Br pX8 qq ` d‹GHP pBr pYn q, Br pY8 qq . By assumption, the right hand side tends to 0 as n Ñ 8. The result then follows from the definition of dLGHP .



As discussed in Section 4.1, all graphs are endowed with edge lengths and viewed ´ ¯1{4 as length spaces. Recall that kn “ 40¨rpnq , and the following notations from 21 Sections 4.1.1 and 4.1.3: given Q P Qn , ‚ Λ “ Pp1q pQq is the largest submap pendant to RpQq, ´ ¯ p ‚ ΛpQq “ Λ, k1n ¨ dΛ , ρQ , 9k84 ¨ νΛ , n

`

`

‚ R “ R pQq “ Q ´ vpΛqztρQ u, ´ ¯ 1 1 ` ` p ` ` ‚ R pQq “ R , kn ¨ dR , ρQ , |vpR` q| ¨ νR . The following lemma relies on Proposition 4.11.2. rpnq

Lemma 4.9.2. For Qn Pu Qn , ´ ¯ d ` p p R pQn q, ΛpQn q Ñ pm8 , Pq

4.9. PROOFS OF THE MAIN THEOREMS

129

as n Ñ 8 for the local Gromov-Hausdorff-Prokhorov topology, where m8 and P are independent.

d p ` pQn q Ñ Proof. By Proposition 4.8.2, R m8 for the pointed GHP topology,

hence also holds for the local GHP topology. Moreover, we show in Proposition 4.11.2 d p nq Ñ that ΛpQ P for the local GHP topology. Finally, the independence between

m8 and P follows from the conditional independence of R` pQn qztρQn u and Pp1q pQn q given that the size of the root block is rpnq.



Proof of Theorem 4.1.1. It follows from Lemma 4.9.2 and the Skorokhod rep´ ¯ p ` pQn q, ΛpQ p nq Ñ resentation theorem that there exists a probability space where R pm8 , Pq almost surely. Lemma 4.9.1 then yields that in this space we have ´ ¯ p ` pQn q, ΛpQ p n q Ñ Z pm8 , Pq Z R almost surely, which implies convergence in distribution. It is easily seen that ´ ¯ p ` pQn q, ΛpQ p nq Z R ˆ ˙ 1 8 1 d “ Qn , ¨ dQn , ρQn , ¨ νΛpQn q ` ¨ pνR` pQn q ´ 1q . kn 9kn4 |vpR` pQn qq| In the above, pνR` pQn q ´ 1q can be replaced by νR` pQn q without affecting the convergence in distribution since |vpR` pQn qq| Ñ 8 as n Ñ 8. Finally, from the definitions of S, P, and m8 , given in Sections 4.10, 1.2.1, d

and 1.1.3, respectively, we have Z pm8 , Pq “ S. Briefly: the equivalence of metric structure is clear, and the measure of S, defined as pπ1 ˝pp1q q˚ Lebr0,1s `pπ8 ˝p8 q˚ LebR , is the same as the measure of Z pm8 , Pq, since the measure of the point in m8 which is glued to the distinguished point of P is almost surely 0. This concludes the proof.



Theorem 4.1.2 follows from Proposition 4.8.1 in the same way as Theorem 4.1.1 follows from Proposition 4.8.2, so we omit the proof.

4.10. THE BROWNIAN PLANE WITH MINBUS

130

4.10. The Brownian Plane with Minbus For s, t P R, we write s ^ t “ minps, tq and s _ t “ maxps, tq. Let e “ pet qtPr0,1s be a standard Brownian excursion. Define a process Z 1 “ pZt1 qtPr0,1s such that, conditioned on e, Z 1 is a centred Gaussian process with covariance E rZs1 Zt1 | es “

min rPrs^t,s_ts

er

for any s, t P r0, 1s. Then shift the time index of the pair pet , Zt1 qtPr0,1s so that the “new Z01 ” is minimal among pZt1 qtPr0,1s . More precisely, by [51, Proposition 2.5], there exists an almost surely unique time s˚ P r0, 1s such that Zs1 ˚ “ mintZt1 : t P r0, 1su. Then for any t P r0, 1s, let (4.10.1)

¯t “ es˚ ` es˚ ‘t ´ 2 e

inf rs˚ ^s˚ ‘t,s˚ _s˚ ‘ts

er ,

Z¯t1 “ Zs1 ˚ ‘t ´ Zs1 ˚ , where s˚ ‘ t “ s ` t if s˚ ` t ď 1, and s˚ ‘ t “ s ` t ´ 1 otherwise. By [51, Theorem 1.2], p¯ et , Z¯t1 qtPr0,1s has the same distribution as pet , Zt1 qtPr0,1s conditioned on ¯ may be viewed mintPr0,1s Zt1 ě 0. The Continuum Random Tree (CRT) coded by e as the CRT coded by e re-rooted at the vertex with minimal label, and the labels ¯ are derived from Z 1 by subtracting the minimal label; Z¯ 1 on the CRT coded by e see Beltran & Le Gall [17, Section 2.3]. Next, let R “ pRt qtě0 and R1 “ pRt1 qtě0 be two independent three-dimensional Bessel processes started from 0, independent of e. We define R “ pRt qtPR by setting $ ’ ’ &Rt if t ě 0 (4.10.2) Rt “ . ’ ’ 1 %R´t if t ă 0 Then for any s, t P R, let $ ’ ’ &rs ^ t, s _ ts st “ ’ ’ %p´8, s ^ ts Y rs _ t, 8q

if st ě 0 , if st ă 0

4.10. THE BROWNIAN PLANE WITH MINBUS

131

and define a process Z “ pZt qtPR such that, conditioned on R, Z is the centred Gaussian process with covariance (4.10.3)

E rZs Zt | Rs “ inf Rr rPst

for any s, t P R. ¯ and R by setting Let X “ pXt qtPR be a concatenation of e $ ’ ’ ¯t e ’ ’ ’ & Xt “ Rt´1 ’ ’ ’ ’ ’ %R1 ´t

if 0 ď t ď 1 if t ą 1

.

if t ă 0

Similarly, let W “ pWt qtPR be given by $ ’ ’ Z¯t1 ’ ’ ’ & Wt “ Zt´1 ’ ’ ’ ’ ’ %Z t

if 0 ď t ď 1 if t ą 1

.

if t ă 0

For any s, t P R, define $ ’ ’ &p´8, s ^ ts Y rs _ t, 8q p st “ ’ ’ %rs ^ t, s _ ts

if st ă 0 and s _ t ą 1 . otherwise

Now, we define a random pseudo-metric dX on R2 by setting, for any s, t P R, dX ps, tq “ Xs ` Xt ´ 2 inf Xr . p rPst

Write s „X t if dX ps, tq “ 0, and let T “ R{ „X . Informally, we may view T as obtained from gluing the root of Aldous’ CRT at the root of infinite Brownian tree. It is easily seen that W0 “ W1 “ 0, E rpWs ´ Wt q2 | Xs “ dX ps, tq, and W has a modification with continuous paths (we shall view W as such in the sequel). Then dX ps, tq “ 0 implies Ws “ Wt almost surely, so we may view W as indexed by T , and we do so.

4.11. CONVERGENCE TO THE BROWNIAN PLANE IN THE GROMOV-HAUSDORFF-PROKHOROV TOPOLOGY For any s, t P R, let (4.10.4)

D˝ ps, tq “ Ws ` Wt ´ 2

inf rPrs^t,s_ts

Wr .

Write p : R Ñ T for the canonical projection, then we extend the domain of D˝ to T ˆ T by setting, for any a, b P T , D˝ pa, bq “ min tD˝ ps, tq : s, t P R, ppsq “ a, pptq “ bu . Let (4.10.5)

D˚ pa, bq “

inf

a0 “a,a1 ,...,ak “b

k ÿ

D˝ pai´1 , ai q

i“1

with the infimum taken over all choices of k P N and of the finite sequence a0 “ a, a1 , . . . , ak “ b P T . It follows that D˚ is a pseudo-metric on T . Write S “ T {tD˚ “ 0u, and let ρ P S be the equivalence class of pp0q. Finally, write π for the canonical projection from T to S, write LebI for the Lebesgue measure over interval I Ă R, and let µ “ pπ ˝ pq˚ LebR . The pointed measured Brownian plane with minbus is the pointed measured metric space S :“ pS, D˚ , ρ, µq. 4.11. Convergence to the Brownian Plane in the Gromov-Hausdorff-Prokhorov Topology In this section, we establish the convergence towards the Brownian plane in the Gromov-Hausdorff-Prokhorov (GHP) topology, extending the result of [30] for the Gromov-Hausdorff (GH) topology. 4.11.1. Scaled Brownian Map. We elaborate a bit on the definition of the scaled Brownian map [30], to make this chapter more self-contained, but follow the notation of that paper. See [30, Section 2.2] for omitted definitions. Fix λ ą 0 in this subsection, and let eλ “ peλt qtPr0,λ4 s be a Brownian excursion of lifetime λ4 . Write Tpλq for the scaled Brownian Continuum Random Tree (CRT) indexed by eλ , and let ppλq : r0, λ4 s Ñ Tpλq be the canonical projection, sending

132

4.11. CONVERGENCE TO THE BROWNIAN PLANE IN THE GROMOV-HAUSDORFF-PROKHOROV TOPOLOGY

133

x P r0, λ4 s to its equivalence class in Tpλq . Conditionally given eλ , let Z λ “ pZtλ q0ďtďλ4 be the centred Gaussian process with covariance ˇ ‰ “ E Zsλ Ztλ ˇ eλ “

(4.11.1)

min rPrs^t,s_ts

eλr .

Furthermore, for s, t P r0, λ4 s with s ď t, we let ˆ Dλ˝ ps, tq



Dλ˝ pt, sq



Zsλ

`

Ztλ

´ 2 max

˙ min rPrs,ts

Zrλ ,

min

rPrt,λ4 sYr0,ss

Zrλ

.

Now extend the domain of Dλ˝ to Tpλq ˆ Tpλq by setting, for any a, b P Tpλq , Dλ˝ pa, bq “ min Dλ˝ ps, tq : s, t P r0, λ4 s, ppλq psq “ a, ppλq ptq “ b

(

,

and Dλ˚ pa, bq



p ÿ

inf

a0 “a,a1 ,...,ap “b

Dλ˝ pai´1 , ai q ,

i“1

where the infimum is over all choices of the integer p ě 1 and of the finite sequence a0 , . . . , ap in Tpλq such that a0 “ a and ap “ b. It follows that Dλ˚ is a pseudo-metric on Tpλq . Write Y λ “ Tpλq {tDλ˚ “ 0u, and let ρλ P Y λ be the equivalence class in Y λ of ppλq p0q. Finally, write πλ for the canonical projection from Tpλq to Y λ , and let µλ “ pπλ ˝ ppλq q˚ Lebr0,λ4 s . The pointed measured scaled Brownian map is Yλ :“ pY λ , Dλ˚ , ρλ , µλ q. Taking λ “ 1, Tp1q is the Brownian CRT, Y 1 “ M, and m8 “ pY 1 , D1˚ , ρ1 , µ1 q is the pointed measured Brownian map. For any λ ą 0, write λ ¨ m8 “ pY 1 , λ ¨ D1˚ , ρ1 , λ4 ¨ µ1 q . Fact 4.11.1. For all λ ą 0, d

λ ¨ m8 “ Yλ . Proof. Fix λ ą 0.

d

As remarked in [30, (5)], we have pY 1 , λ ¨ D1˚ , ρ1 q “ d

pY λ , Dλ˚ , ρλ q. Furthermore, the equality λ4 ¨ µ1 “ µλ follows from the fact that

4.11. CONVERGENCE TO THE BROWNIAN PLANE IN THE GROMOV-HAUSDORFF-PROKHOROV TOPOLOGY

134

they are both push-forwards of Lebr0,λ4 s by two functions which are equivalent in distribution.



4.11.2. A Nice Event. [30, Proposition 4] defines an event on which, λ ¨ m8 and P have the same local metric structure. In Proposition 4.11.2 below, we show that on this event, λ ¨ m8 and P also have the same local structure with their endowed measures. The purpose of the current subsection is to describe this event. Fix A ą 1, α ą 0 and λ ą p2αq1{4 . Let eλ be a copy of Brownian excursion of lifetime λ4 , and let R “ pR, R1 q, independent of eλ , be copies of independent threedimensional Bessel processes. Next, let Z and Z λ be centred Gaussian processes with covariances, respectively, given in (4.10.3) and (4.11.1). Furthermore, for every x ě 0, let γ8 pxq “ suppt ě 0 : Rt “ xq . Now define the event Eλ “ EA,α,λ peλ , R, R1 q " “

eλt

“ Rt and

eλλ4 ´t



Rt1 , @t

(

ďα X

* min

αďtďλ4 ´α

eλt

“ inf Rt ^ inf těα

těα

Rt1

.

As in the proof of [5, Proposition 4], on Eλ we have Ztλ “ Zt , Zλλ4 ´t “ Z´t , @t P r0, αs . Then let Fλ “ FA,α,λ peλ , R, R1 , Z λ , Zq be the intersection of Eλ with the following events: inf Rt ^ inf Rt1 ą A4

těα

těα

and min Zγ8 pxq ă ´10 ,

0ďxďA

min Zγ8 pxq ă ´10 ,

AďxďA2

min Zγ8 pxq ă ´10 .

A2 ďxďA4

4.11.3. Convergence to the Brownian Plane. Recall from Section 4.1 that, ř given Q “ pQ, uvq P Q, we let νQ “ vPvpQq δv . Since local GHP convergence is only

4.11. CONVERGENCE TO THE BROWNIAN PLANE IN THE GROMOV-HAUSDORFF-PROKHOROV TOPOLOGY stated for length spaces, we view each edge e of Q as an isometric copy of the unit interval r0, 1s. We abuse notation and continue to write pQ, dQ q for the resulting length space. In this section, for c ą 0 we write ˆ c¨Q“

8c4 Q, c ¨ dQ , u, ¨ νQ 9

˙ .

In the sequel, given Q “ pQ, uvq P Q, we always let u be the distinguished point of Q. Proposition 4.11.2. Let pkn P R` : n P Nq be such that kn Ñ 8 and kn “ opn1{4 q as n Ñ 8. Then for Qn Pu Qn , kn´1 ¨ Qn Ñ P in distribution for the local Gromov-Hausdorff-Prokhorov topology. Recall that given a pointed measured metric space V “ pV, d, o, νq, writing Br “ Br pVq, we let (4.11.2)

´ ˇ ¯ Br pVq “ pBr , d, oq and Br pVq “ Br , d, o, ν ˇBr ,

ˇ where ν ˇBr denotes the measure ν restricted to Br . By the definition of local GHP, it suffices to show that, given Qn Pu Qn , for any r ě 0, (4.11.3)

d

Br pkn´1 ¨ Qn q Ñ Br pPq

as n Ñ 8 for the pointed GHP topology; see Sections 4.2.1 and 4.2.2 for definitions of pointed GHP and local GHP topologies, respectively. We will show the convergence for r “ 1, for ease of notation, and the argument for r ‰ 1 follows similarly. Proof of Proposition 4.11.2. This proof is a slight extension of the proof for [30, Theorem 1.2]. Fix ε ą 0. It follows immediately from [30, Proposition 3] and the proof of [30, Proposition 4] that there exists A ą 1, α ą 0, and λ0 ą p2αq1{4 such that

135

4.11. CONVERGENCE TO THE BROWNIAN PLANE IN THE GROMOV-HAUSDORFF-PROKHOROV TOPOLOGY for all λ ě λ0 , we can construct copies of eλ , R, R1 , Z λ , Z, λ ¨ m8 , and P on a common probability space in such a way that, with probability at least 1 ´ ε, the event Fλ “ FA,α,λ peλ , R, R1 , Z λ , Zq holds. As shown in [30, Proposition 4], on the event Fλ , it holds that B1 pλ ¨ m8 q “ B1 pPq . On the other hand, by [30, Proposition 9], there exists α0 “ α0 pεq ą 0 such that, for every sufficiently large integers m and n with n ą m, we can construct Qn Pu Qn and Qm Pu Qm on a common probability space in such a way that the equality Bα0 m1{4 pQn q “ Bα0 m1{4 pQm q holds with probability at least 1 ´ ε. Without loss of generality, we assume that α0 ă n P N. We write λ“

α0´1

1 2λ0

and kn ď α0 tn1{4 u for all

ˆ ˙1{4 8 , 9

and note that λ ą λ0 . For n P N, let mn “ rα0´1 kn s4 . Since mn tends to infinity with n, it follows that for large enough n, we may couple Qn and Qmn such that the equality (4.11.4)

B1 pkn´1 ¨ Qn q “ B1 pkn´1 ¨ Qmn q

holds with probability at least 1´ε. Since νQn and νQmn both are counting measures, it follows from (4.11.4) that (4.11.5)

ˇ ˇ νQn ˇB1 pkn´1 ¨Qn q “ νQmn ˇB1 pkn´1 ¨Qm

with probability at least 1 ´ ε. In the remainder of the proof, we let T “ λ4 .

nq

136

4.11. CONVERGENCE TO THE BROWNIAN PLANE IN THE GROMOV-HAUSDORFF-PROKHOROV TOPOLOGY

137

Next, for every x P r0, eλT {2 s, we set ` ˘ ` ˘ γλ pxq “ sup t ď T {2 : eλt “ x , ηλ pxq “ inf t ě T {2 : eλt “ x . By [30, Lemma 5], on the event Fλ , if Dλ˚ pρλ , ppλq ptqq ď 1 then t P r0, γλ pAqq Y pηλ pAq, T s. From the proof of [30, Proposition 4], we also know that γλ pAq ă α and T ´ ηλ pAq ă α. Recalling the definition of Fλ from Section 4.11.2, it follows that on Fλ , we simultaneously have eλt “ Rt , Ztλ “ Zt , @t P r0, γλ pAqs , and eλt “ RT1 ´t , Ztλ “ Zt´T , @t P rηλ pAq, T s . This implies that on Fλ , ˇ πλ ˝ ppλq ˇr0,γ

λ pAqqYpηλ pAq,T s

ˇ “ π8 ˝ p8 ˇr0,γ

λ pAqqYpηλ pAq´T,0s

,

where πλ ˝ ppλq is the canonical projection from r0, T s to λ ¨ m8 , and π8 ˝ p8 is the canonical projection from R to P. Since B1 pλ¨m8 q Ă πλ ˝ppλq pr0, γλ pAqq Y pηλ pAq, T sq and B1 pPq Ă π8 ˝ p8 pr0, γλ pAqq Y pηλ pAq ´ T, 0sq, we obtain that, on Fλ , the measured versions of B1 pλ ¨ m8 q and B1 pPq are also equal: (4.11.6) Next, since

B1 pλ ¨ m8 q “ B1 pPq . 8mn 4 9kn

“ λ4 for all n P N, it follows from [59, Theorem 1] and [48,

Theorem 1.1] that d

kn´1 ¨ Qmn Ñ λ ¨ m8 for the pointed GHP topology. (In [30, 59], the convergence is only stated for the GH topology, but the proof in fact yields the above formulation. This is also stated explicitly in [4, Theorem 4.1].) So (4.11.7)

d

B1 pkn´1 ¨ Qmn q Ñ B1 pλ ¨ m8 q

4.12. EXPECTED SIZE OF PENDANT SUBMAP

for the pointed GHP topology. Finally, it follows that we may simultaneously couple Qn and Qmn , λ ¨ m8 , and P so that with probability at least 1 ´ 2ε we have both B1 pkn´1 ¨ Qn q “ B1 pkn´1 ¨ Qmn q and B1 pλ ¨ m8 q “ B1 pPq . In a space where such a coupling holds, for any bounded continuous function F : K‹ Ñ R, we have ˇ “ ˇ “ ‰ˇ ‰ˇ ˇE F pB1 pkn´1 ¨ Qn qq ´ F pB1 pPqq ˇ ď ˇE F pB1 pkn´1 ¨ Qn qq ´ F pB1 pkn´1 ¨ Qmn qq ˇ ˇ “ ‰ˇ ` ˇE F pB1 pkn´1 ¨ Qmn qq ´ F pB1 pλ ¨ m8 qq ˇ ` |E rF pB1 pλ ¨ m8 qq ´ F pB1 pPqqs| . Writing }F } “ supxPK‹ F pxq, the first and the third terms on the right of the inequality are each less than 2ε}F }. The second term tends to 0 with n, by (4.11.7). Therefore, ˇ “ ˇ ‰ lim sup ˇE F pB1 pkn´1 ¨ Qn qq ´ E rF pB1 pPqqsˇ ă 4ε}F } . nÑ8

Since ε was arbitrary, it follows that E rF pB1 pkn´1 ¨ Qn qqs Ñ E rF pB1 pPqqs, so d

B1 pkn´1 ¨ Qn q Ñ B1 pPq for the pointed GHP topology by the Portmanteau theorem. As noted above, the case r ‰ 1 of (4.11.3) follows by a similar argument. 

4.12. Expected Size of Pendant Submap In this section, we derive the value of ν “ E rξs. Recall from (4.6.1) that for k P N, ř ` ` P pξ “ kq “ 2?3 π 1`ψpkq . Since |Q``2 | “ ?2π 12 p1`ψp``1qq for ` P N, 8 `“0 |Q``2 | ¨ z “ k5{2 p``1q5{2 1 1 ` M pzq, and M p 12 q “ 31 , we have 8 ÿ

E rξs “ k“1

k ¨ P pξ “ kq

138

4.12. EXPECTED SIZE OF PENDANT SUBMAP



8 3ÿ 2 1 ` ψpkq k¨? 4 k“1 π k 5{2

8 3ÿ 1 “ p` ` 1q ¨ |Q``2 | ¨ ` 4 `“0 12 ˜ ¸ 8 1 3 ÿ 4 ` ¨ |Q``2 | ¨ ` ` “ 4 `“1 12 3 ˆ ˙ 3 1 1 1 “ ¨ M `1 4 12 12



1 M 1 p 12 q `1 . 16

1 To compute M 1 p 12 q, we use the following formula from [16, (10)]:

M pzq “ ´1 `

˘ 1 ` ´1 ` 18z ` p1 ´ 12zq3{2 . 2 54z

1 Then M 1 p 12 q “ 16. It follows that

ν “ E rξs “ 2 .

139

Conclusions

141

CHAPTER 5

Conclusions Summaries of the main results and methodologies are presented in Sections 5.1 and 5.2, respectively.

5.1. Summary of Main Results In Theorems 3.1.2 and 3.1.3 of Chapter 3, we establish two versions of the following joint convergence: a uniform rooted quadrangulation with n vertices, its largest 2-connected block, and its largest simple block, after rescaling the graph ` 9 ˘1{4 distance by 8n , converge to the same Brownian map for the Gromov-HausdorffProkhorov topology. Theorem 3.1.3 conditions on the sizes of the largest 2-connected block and of the largest simple block, while in Theorem 3.1.2 we consider a uniform rooted quadrangulation without conditioning on block sizes. Theorem 3.1.2 thus indicates the convergence of rooted 2-connected quadrangulations with random sizes. Subsequently, in Theorem 3.1.1, we deduce the convergence of a uniform ` 21 ˘1{4 rooted 2-connected quadrangulation with r vertices, after rescaling by 40r , to the Brownian map. Built upon the convergence result for rooted 2-connected quadrangulations, we further obtain the subcritical scaling limit for rooted quadrangulations conditional on having an exceptionally large 2-connected root block. That is, if r : N Ñ N satisfies that rpnq “ opnq as n Ñ 8 and rpnq ą pln nq25 for all n, then a uniform rooted quadrangulation with n vertices conditional on having a 2-connected root ´ ¯1{4 21 , converges to the Brownian plane block with size rpnq, after rescaling by 40rpnq with the Brownian map attached at the distinguished point, for the local GromovHausdorff-Prokhorov Topology. This is the content of Theorems 4.1.1 and 4.1.2.

5.2. SUMMARY OF METHODOLOGIES

These convergence results provide further evidence to the universality of the Brownian map and the Brownian plane, and introduce an intriguing new limit which shows it is possible to “zoom in” on the minimal neck baby universes hiding within the Brownian plane. 5.2. Summary of Methodologies In Chapter 3, we use generating functions to describe map decompositions, followed by using singularity analysis to deduce the asymptotic sizes of some families of maps. Both techniques are mainly inspired by the work of Banderier, Flajolet, Schaeffer & Soria [16]. These methodologies are suitable when we concern about combinatorial structures of maps expressible by recursive compositional schemata, and they often provide useful characterizations for the size distributions of various map components. We also use the concept of uniformly asymptotically negligible attachments in both Chapters 3 and 4. Briefly speaking, given a uniform rooted quadrangulation, upon removing the largest (2-connected or simple) block, the rest of the components exhibit exchangeability and they attach to the largest block uniformly; furthermore, their sizes are asymptotically negligible. With such property at hand, we can ensure that the attachments do not concentrate on a small region of the largest block. This is what allows the convergence in the Gromov-Hausdorff topology to be strengthened to the convergence in the Gromov-Hausdorff-Prokhorov topology.

142

Potential Future Works

144

CHAPTER 6

Potential Future Works Potential future works related to this thesis are highlighted here. In Section 6.1 we discuss concentration of the total mass of small submaps pendant to the 2connected root block of a quadrangulation when this block is conditioned to be exceptionally large. In Section 6.2, we consider the scaling limit of 3-connected general maps. Finally, in Section 6.3 we consider properties of large planar graphs.

6.1. Concentration of the Total Mass of Non-Largest Pendant Submaps In this section, when we say block we mean 2-connected block. rpnq

Recall that Qn

denotes the set of rooted quadrangulations of size n conditional

on the event that the root block has rpnq vertices. In this subsection, for each n P N rpnq

let Qn Pu Qn . Recall that νG “

ř

vPvpGq δv

denotes counting measure on the vertices of a graph

G. In Theorem 4.1.1, the measure assigned to Qn is 7 1 ¨ νΛpQn q ` ¨ νR` pQn q , ` 15rpnq |vpR pQn qq| recalling relevant notations from Section 4.1.1. We conjecture that replacing this measure by

7 15rpnq

¨ νQn does not affect the scaling limit result. We formalize this

conjecture as follows. Conjecture 6.1.1. Let r : N Ñ N be such that rpnq Ñ 8 and rpnq “ opnq as n Ñ 8. Then for Qn Pu Qn,rpnq , we have ˜

ˆ Qn ,

21 40rpnq

˙1{4

7 ¨ dQn , ρQn , ¨ νQ n 15rpnq

¸ ÑS

in distribution for the local Gromov-Hausdorff-Prokhorov topology.

6.1. CONCENTRATION OF THE TOTAL MASS OF NON-LARGEST PENDANT SUBMAPS 145 The above scaling factor for the metric is the same as that for Theorem 4.1.1. Briefly, since the root block of Qn has size rpnq, it follows from Theorem 3.1.1 that, ´ ¯1{4 21 upon rescaling the graph distance by 40rpnq , the root block converges to the Brownian map. The above scaling factor for the measure will follow from Theorem 4.1.1, at least in the case that rpnq ą pln nq25 for all n, upon establishing the next conjecture. Conjecture 6.1.2. Let r : N Ñ N be such that rpnq Ñ 8 and rpnq “ opnq as n Ñ 8. Then for Qn Pu Qn,rpnq , we have ˇ ˇ ˇ ˇ ˇ|vpR` pQn qq| ´ 15rpnq ˇ “ oprpnqq ˇ 7 ˇ with probability 1 ´ op1q. Proof of Conjecture 6.1.1 assuming Conjecture 6.1.2. Fix ε ą 0 and n P N. Let Qn Pu Qn,rpnq . Recall that Qn can be viewed as the metric space obtain by gluing ΛpQn q to R` pQn q at a point. Together with the discussions succeeding Conjecture 6.1.1, for each n P N, the GHP distance between the pointed measured metric spaces in the statements of Conjecture 6.1.1 and Theorem 4.1.1 is bounded by ˆ

7 7 1 dP ¨ νQn , ¨ νΛpQn q ` ¨ νR` pQn q ` 15rpnq 15rpnq |vpR pQn qq| ˆ ˙ 1 7 ď dP ¨ νR` pQn q , ¨ νR` pQn q , 15rpnq |vpR` pQn qq| ´ where dP is the L´evy-Prokhorov distance on

21 40rpnq

˙

¯1{4 ¨ Qn .

Now we turn to bounding the above Prokhorov distance. Fix V Ă vpQn q. Assume that Conjecture 6.1.2 is proven, then there is h : N Ñ R with hpnq “ op1q as n Ñ 8 ˇ ˇ ˇ ˇ such that with probability 1 ´ op1q we have ˇ|vpR` pQn qq| ´ 15rpnq ď hpnqrpnq. Since 7 ˇ νR` pQn q is counting measure over the vertices of R` pQn q, it follows that ˇ ˆˇ ˙ ˇ 7 ˇ 1 ˇ ˇ P ˇ ¨ νR` pQn q pV q ´ ¨ νR` pQn q pV qˇ ą ε 15rpnq |vpR` pQn qq|

6.1. CONCENTRATION OF THE TOTAL MASS OF NON-LARGEST PENDANT SUBMAPS 146 ˆ ďP

7|hpnq| ¨ νR` pQn q pV q ą ε 15|vpR` pQn qq|

˙

Ñ0 as n Ñ 8. Finally, using a similar argument as in Corollary 3.7.2, we obtain ˆ dP

1 7 ¨ νR` pQn q , ¨ νR` pQn q ` 15rpnq |vpR pQn qq|

˙ Ñ0.

Conjecture 6.1.1 then follows.



Next, we describe two ideas for proving Conjecture 6.1.2.

6.1.1. Idea One. We aim to apply a result of Armend´ariz & Loulakis [15] to show that sizes of submaps of R` pQn q pendant to RpQn q are asymptotically independent with high probability, and thereby obtain concentration of |vpR` pQn qq|. (1) Asymptotic independence. We start by introducing notation from [15]. Let T :

Ť nPN

Rn Ñ

Ť nPN

Rn be the

operator that exchanges the last and the maximum component of a finite sequence: $ ’ ’ max1ďiďn xi ’ ’ ’ & T px1 , . . . , xn qk “ xn ’ ’ ’ ’ ’ %x k

if k “ n if xk ě max1ďiăk xi , xk “ maxiěk xi . if otherwise

Next, fix t ą 0, let ∆ “ p0, ts, and denote by x`∆ the interval px, x`ts. Let µ be a probability measure on R, let X1 , X2 , . . . be independent variables with law µ, and ř write Sn “ nk“1 Xk for n P N. We say that µ is ∆-subexponential if µpx ` ∆q ą 0 for all large enough x, and if we simultaneously have µpx ` y ` ∆q “ 1 @y P R , xÑ8 µpx ` ∆q lim

P pSn P x ` ∆q “ 1 @n P N . xÑ8 n ¨ µpx ` ∆q lim

6.1. CONCENTRATION OF THE TOTAL MASS OF NON-LARGEST PENDANT SUBMAPS 147 An immediate consequence of the last asymptotics is ˇ ˇ ˇ ˇ P pSn P x ` ∆q ˇ ´ 1ˇˇ “ 0 , lim sup ˇ nÑ8 xědn n ¨ µpx ` ∆q

(6.1.1)

for an appropriate sequence pdn : n P Nq tending to infinity with n; see [15, 33] for sufficient conditions on dn . Furthermore, for n P N and x P R, let ˇ ` ˘ ˇ µ∆ n,x pAq “ P pX1 , . . . , Xn q P A Sn P x ` ∆ for any Borel set A Ă Rn . The paper Armend´ariz & Loulakis [15] derives the asymptotic behaviour of µ∆ n,x when n Ñ 8 and x ě dn ; the result we require can be stated as follows. Theorem 6.1.1 ([15, Theorem 1]) Suppose µ is ∆-subexponential. There exists a sequence qn such that sup

lim sup

nÑ8 xěqn APBpRn´1 q

ˇ ∆ ˇ ˇµn,x ˝ T ´1 pA ˆ Rq ´ µn´1 pAqˇ “ 0 ,

where BpRn´1 q denotes the set of Borel sets of Rn´1 .

The theorem says that, under (6.1.1), the condition Sn P x ` ∆ only affects the maximum of X1 , . . . , Xn in the limit, and the n ´ 1 remaining variables are asymptotically independent. Now, let µ be the law of ξ, as defined in (4.4.2).

It follows that µ is ∆-

subexponential. Recall that ξ1 , ξ2 , . . . are independent copies of ξ. Write Sk “ řk i“1 ξi . For each n P N, let Qn Pu Qn,rpnq , and write N pnq “ |FpQn q| for the number of facial 2-cycles in the pre-root-block of Qn . It follows that, given n, k P N, for A P BpRk q, `` P

ˇ ˘ ˘ Y1 pQn q, . . . , YN pnq pQn q P A ˇ N pnq “ k

ˇ ` ˘ “ P pξ1 , . . . , ξk q P A ˇ Sk “ n ´ rpnq p0,1s

“ µk,n´rpnq´1 pAq .

6.1. CONCENTRATION OF THE TOTAL MASS OF NON-LARGEST PENDANT SUBMAPS 148 We want to apply Theorem 6.1.1 to show that, with high probability, (6.1.2)

lim

sup

nÑ8 APBpRN pnq´1 q

ˇ ˇ ˇ p0,1s ˇ ˇµN pnq,n´rpnq´1 ˝ T ´1 pA ˆ Rq ´ µN pnq´1 pAqˇ “ 0 .

To achieve that, we need to show that, with high probability, the following hold simultaneously: (1) N pnq Ñ 8 as n Ñ 8; and (2) The sequence pqn : n P Nq can be chosen such that n ´ rpnq ě qn for all but finitely many n, where qn is the sequence from Theorem 6.1.1. We quickly show (1) below, under the assumption that rpnq ą C ln n for C sufficiently large. The proof of (2) requires deriving the precise conditions on qn under which Theorem 6.1.1 is valid. We expect to return to this in future work. To obtain a lower bound for N pnq, we use a similar argument as for Lemma 4.6.4 and Proposition 4.6.1. We only describe the differences in the proof, as follows. Let n, r, k P N and ε P p0, 47 q with k ď

4r 7

7

´ εr and n ą p2ν ` 1qr. We have

4r 7

ˆ |Qn,r,k | ď |Qn,r,r 4r s | ¨

4r 7

´ εr 9 ¨ ´ εr ` 2r 2

˙r 4r7 s´k ¨

P pSk “ n ´ rq ´ ¯ P Sr 4r s “ n ´ r 7

ˆ ď |Qn,r,r 4r s | ¨ 1 ´ 7

49ε 36

˙r 4r7 s´k

49ε 36

˙r 4r7 s´k

¨

P pSk “ n ´ rq ´ ¯ P Sr 4r s “ n ´ r 7

ˆ ď |Qn,r,r 4r s | ¨ 1 ´ 7

n5{2 , δ

for some δ ą 0. It follows that for Qn Pu Qn,rpnq , taking ε “ 71 , ˆ

4rpnq rpnq ´ P |FpQn q| ď 7 7

˙ ÿ ď 1ďkď 4rpnq ´ rpnq 7 7

|Qn,rpnq,k | |Qn,rpnq,r 4rpnq s | 7

ˆ ÿ ď 1ďkď 4rpnq ´ rpnq 7 7

36 ď 7δ

ˆ

7 1´ 36

7 1´ 36

˙ rpnq 7

˙r 4rpnq s´k 7

n5{2 .

n5{2 δ

6.1. CONCENTRATION OF THE TOTAL MASS OF NON-LARGEST PENDANT SUBMAPS 149 Recalling that N pnq “ |FpQn q| and rpnq ą C ln n for sufficiently large C, we thus have N pnq ą

3rpnq 7

with 1 ´ op1q probability as n Ñ 8. This shows that N pnq Ñ 8

as n Ñ 8 with high probability.

(2) Concentration. Now we turn to showing concentration of |vpR` pQn qq|. Note that `

|vpR pQn qq| “ rpnq `

Nÿ pnq

Ypiq pQn q .

i“2

Assuming that (6.1.2) holds, the law of |vpR` pQn qq| is close to rpnq`

řN pnq´1 i“1

ξi , with

high probability as n Ñ 8. Since ξi ’s are asymptotically 23 -stable random variables, ř pnq´1 ξi fluctuates around its mean with order N pnq2{3 . Since their sum SN pnq´1 “ N i“1 N pnq Ñ 8 in probability, it follows that ˇ `ˇ ˘ lim P ˇSN pnq´1 ´ νN pnqˇ ă N pnq3{4 “ 1 .

nÑ8

Furthermore, fix a sufficiently small ε ą 0. Using the same logic as above, we deduce that ˇ ˆˇ ˙ ˆ ˙εrpnq ˇ ˇ 4rpnq 72 49ε ˇ ě εrpnq ď P ˇˇN pnq ´ 1´ n5{2 . 7 ˇ 49εδ 36 It thus follows that ˆ

˙ ˇ ˇ 1 ˇ ˇ P SN pnq´1 ´ νN pnq ě εrpnq 2 ˇ ˇ ˆ ˙ ˇ ˇ 1 ˇ ˇ 4rpnq ˇ ˇ ˇ ˇ ď P SN pnq´1 ´ νN pnq ě εrpnq, ˇN pnq ´ ă εrpnq 2 7 ˇ ˇ ˆˇ ˙ ˇ ˇ 4rpnq ˇ ě εrpnq ` P ˇˇN pnq ´ 7 ˇ ˇ ˆ ˙ ˆˇ ˙ ˇ ˇ ˇ ˇ N pnq 4rpnq ˇ ě εrpnq ď P ˇSN pnq´1 ´ νN pnqˇ ą 8 ` P ˇˇN pnq ´ ˇ 7 ` 2 7ε Ñ0

6.1. CONCENTRATION OF THE TOTAL MASS OF NON-LARGEST PENDANT SUBMAPS 150 as n Ñ 8, provided rpnq ą C ln n for C sufficiently large. As derived in Section 4.12, we have ν “ 2. Then the preceding bound gives ˇ ˆˇ ˙ ˇ ˇ 8rpnq ˇ ˇ P ˇSN pnq´1 ´ ě εrpnq 7 ˇ ˇ ˆ ˙ ˆˇ ˙ ˇ ˇ 1 ˇ ˇ 1 4rpnq ˇ ě εrpnq ď P ˇSN pnq´1 ´ 2N pnqˇ ě εrpnq ` P ˇˇN pnq ´ 2 7 ˇ 4 Ñ0 as n Ñ 8. This together with the previous discussion yields that, with high probability, |vpR` pQn qq| “ rpnq ` SN pnq´1 “

15rpnq ` oprpnqq . 7

6.1.2. Idea Two. The second idea for proving Conjecture 6.1.2 is less concrete, but seems as though it would yield a slicker proof if it could be made to work. We start by observing that, given Qq Pu Qq , its largest block has size

7q `Opq 2{3 q 15

with high probability. To wit, let λ : N Ñ R` be an arbitrarily fixed function such that λpqq Ñ 8 as q Ñ 8. Recall that bpQq q denotes the size of the largest block of Qq , and that A is the Airy density function. It follows from Proposition 3.4.3 that ˙ ˆˇ 7q ˇˇ ˇ 2{3 P ˇbpQq q ´ ˇ ď λpqqq “ 15

7q t 15 `λpqqq 2{3 u

ÿ

P pbpQq q “ kq

7q k“r 15 ´λpqqq 2{3 s

1 ` op1q “ q 2{3

7q t 15 `λpqqq 2{3 u

ÿ

7q k“r 15 ´λpqqq 2{3 s

ż λpqqq2{3 “ p1 ` op1qq

ˆ

k ´ 7q{15 βA β ¨ q 2{3 ˆ

βA β ¨ ´λpqqq 2{3

x q 2{3

˙

˙

dx q 2{3

Ñ1 as q Ñ 8. For Qn Pu Qn,rpnq , its root block RpQn q has size rpnq. Note that any submap of R` pQn q pendent to RpQn q has size at most Yp2q pQn q. By Corollary 4.7.2, with high probability, Yp2q pQn q ď rpnq5{6 , showing that RpQn q is the largest block of R` pQn q.

6.3. PROPERTIES OF LARGE PLANAR GRAPHS

If we could further show that, with high probability, R` pQn q is close in distribution to a uniform quadrangulation, then R` pQn q would necessarily have size of order 15rpnq 7

` oprpnqq with high probability, proving Conjecture 6.1.2.

6.2. The Scaling Limit of 3-Connected General Maps Bettinelli, Jacob & Miermont [23] showed that the Brownian map is the scaling limit of uniform rooted maps with a fixed number of edges. We may use the ideas of this thesis to further deduce the scaling limit of uniform rooted 2-connected and 3-connected maps with a fixed number of edges. The ideas are to decompose general maps recursively into 2-connected and 3-connected blocks; express the combinatorial structures using generating functions; resort to singularity analysis for coefficient extractions; and finally derive the uniformly asymptotically negligible property of submaps pendant to the largest 2-connected or 3-connected block. Recall from Chapter 3 that, with high probability, the largest 2-connected block of a uniform quadrangulation is a “sandwich” of the quadrangulation and the largest simple block, both of which are known to rescale to the Brownian map, so the scaling limit of the largest 2-connected block follows unsurprisingly. However, the scaling limit of general maps with higher connectivity constraints is yet to be established. Hence, we cannot say that the largest 2-connected or 3-connected block of a uniform rooted general map is a “sandwich” of the map and its higher connectivity component. Further efforts are therefore needed to get around this.

6.3. Properties of Large Planar Graphs The Tutte-Whitney theorem says that every 3-connected planar graph has a unique embedding in the plane in which every face is a convex polygon and every edge is a straight line segment, up to orientation-preserving homeomorphism. Once we succeed in deriving the scaling limit of uniform rooted 3-connected general maps, we obtain a tail bound on the diameter. Then using the Tutte-Whitney theorem, we may transfer the diameter bound to the uniform rooted 3-connected planar graphs.

151

6.3. PROPERTIES OF LARGE PLANAR GRAPHS

Using versions of Tutte’s quadratic methods and recursive decompositions, it might then be possible to study the scaling limits of large 2-connected and general planar graphs. Other properties about planar graphs with connectivity constraints may be deduced similarly, such as the asymptotic number of vertices at a given distance from the root (also known as profile), and existence (or non-existence) of the confluence phenomena of geodesics in planar graphs.

152

Bibliography

154

Bibliography [1] Abraham, C. (2013). Rescaled bipartite planar maps converge to the Brownian map. arXiv preprint arXiv:1312.5959. 93 [2] Abraham, R., Delmas, J. F. & Hoscheit, P. (2013). A note on the GromovHausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab, 18(14), 1-21. 95, 99 [3] Addario-Berry, L. (2015). A probabilistic approach to block sizes in random maps. 100, 116 [4] Addario-Berry, L., & Albenque, M. (2013). The scaling limit of random simple triangulations and random simple quadrangulations. arXiv preprint arXiv:1306.5227. 2, 14, 20, 21, 22, 38, 41, 43, 47, 76, 93, 99, 137 [5] Addario-Berry, L. & Wen, Y. (2014). Joint convergence of random quadrangulations and their cores. arXiv preprint arxiv:1503.06738. vi, x, 2, 21, 93, 96, 97, 103, 104, 121, 122, 123, 124, 125, 134 [6] Albenque, M., & Poulalhon, D. (2013). Generic method for bijections between blossoming trees and planar maps. arXiv preprint arXiv:1305.1312. Chicago 34 [7] Aldous, D. J. (1985). Exchangeability and related topics (pp. 1-198). Springer Berlin Heidelberg. 72 [8] Aldous, D. (1991). The continuum random tree. I. The Annals of Probability, 1-28. 4 [9] Aldous, D. (1993). The continuum random tree III. The Annals of Probability, 248-289. 4 [10] Aldous, D. (1993). Tree-based models for random distribution of mass. Journal of Statistical Physics, 73(3-4), 625-641. 4

155

[11] Ambj∅rn, J., & Budd, T. G. (2013). Trees and spatial topology change in causal dynamical triangulations. Journal of Physics A: Mathematical and Theoretical, 46(31), 315201. 14 [12] Ambj∅rn, J., Durhuus, B., & J´onsson, p. (1997). Quantum Geometry: a Statistical Field Theory Approach. Cambridge University Press. 3 [13] Angel, O. (2003). Growth and percolation on the uniform infinite planar triangulation. Geometric & Functional Analysis GAFA, 13(5), 935-974. 15 [14] Angel, O., & Schramm, O. (2003). Uniform infinite planar triangulations. Communications in Mathematical Physics, 241(2), 191-213. 15 [15] Armend´ariz, I., & Loulakis, M. (2011). Conditional distribution of heavy tailed random variables on large deviations of their sum. Stochastic Processes and their Applications, 121(5), 1138-1147. 146, 147 [16] Banderier, C., Flajolet, P., Schaeffer, G., & Soria, M. (2001). Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Structures & Algorithms, 19(3-4), 194-246. iv, vi, 20, 21, 36, 41, 42, 48, 53, 54, 55, 59, 62, 63, 103, 114, 139, 142 [17] Beltran, J., & Le Gall, J. F. (2013). Quadrangulations with no pendant vertices. Bernoulli, 19(4), 1150-1175. 9, 14, 130 [18] Bender, E. A., & Canfield, E. R. (1989). Face sizes of 3-polytopes. Journal of Combinatorial Theory, Series B, 46(1), 58-65. 83, 122 [19] Benjamini, I., & Curien, N. (2013). Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points. Geometric and Functional Analysis, 23(2), 501-531. 15 [20] Bernardi, O., & Fusy, E. (2012). A bijection for triangulations, quadrangulations, pentagulations, etc. Journal of Combinatorial Theory, Series A, 119(1), 218-244. 29 [21] Bettinelli, J. (2012). The topology of scaling limits of positive genus random quadrangulations. The Annals of Probability, 40(5), 1897-1944. 14

156

[22] Bettinelli, J. (2015). Scaling limit of random planar quadrangulations with a boundary. In Annales de l’Institut Henri Poincar´e, Probabilit´es et Statistiques (Vol. 51, No. 2, pp. 432-477). Institut Henri Poincar´e. 14 [23] Bettinelli, J., Jacob, E., & Miermont, G. (2014). The scaling limit of uniform random plane maps, via the Ambj∅rn-Budd bijection. Electron. J. Probab, 19(74), 1-16. 14, 93, 151 [24] Bollob´as, B. (1998). Modern Graph Theory (Vol. 184). Springer Science & Business Media. 25 [25] Bouttier, J., Di Francesco, P., & Guitter, E. (2004). Planar maps as labeled mobiles. Electron. J. Combin, 11(1), R69. 6, 10, 29, 33 [26] Burago, D., Burago, Y. & Ivanov, S. (2001). A Course in Metric Geometry (Vol. 33, pp. 371-374). Providence: American Mathematical Society. 10, 51, 95, 99 [27] Chassaing, P., & Durhuus, B. (2006). Local limit of labeled trees and expected volume growth in a random quadrangulation. The Annals of Probability, 34(3), 879-917. 15 [28] Chassaing, P. & Schaeffer, G. (2004). Random planar lattices and integrated superBrownian excursion. Probability Theory and Related Fields, 128(2), 161212. 5, 6, 49, 66, 118 [29] Cori, R., & Vauquelin, B. (1981). Planar maps are well labeled trees. Canad. J. Math, 33(5), 1023-1042. 29 [30] Curien, N. & Le Gall, J. F. (2012). The Brownian plane. Journal of Theoretical Probability, 1-43. iv, vi, 16, 17, 18, 19, 20, 93, 95, 97, 98, 99, 132, 133, 134, 135, 136, 137 [31] Curien, N., & Le Gall, J. F. (2014). The hull process of the Brownian plane. arXiv preprint arXiv:1409.4026. 16 [32] Curien, N., & Le Gall, J. F. (2014). Scaling limits for the peeling process on random maps. arXiv preprint arXiv:1412.5509. 16

157

[33] Denisov, D., Dieker, A. B., & Shneer, V. (2008). Large deviations for random walks under subexponentiality: the big-jump domain. The Annals of Probability, 36(5), 1946-1991. 147 [34] Even, S. (2011). Graph Algorithms. Cambridge University Press. 28 [35] Fill, J. A., Flajolet, P., & Kapur, N. (2005). Singularity analysis, Hadamard products, and tree recurrences. Journal of Computational and Applied Mathematics, 174(2), 271-313. 36 [36] Flajolet, P., & Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press. 35, 36, 59, 60, 88 [37] Gao, Z., & Wormald, N. C. (1999). The size of the largest components in random maps. SIAM Journal on Discrete Mathematics, 12(2), 217-228. 20, 41, 53 [38] Gouldn, I.P. & Jackson, D.M. (1983). Combinatorial Enumeration. John Wiley & Sons. 58, 59 [39] Gurel-Gurevich, O., & Nachmias, A. (2012). Recurrence of planar graph limits. arXiv preprint arXiv:1206.0707. 15 [40] Jain, S., & Mathur, S. D. (1992). World-sheet geometry and baby universes in 2D quantum gravity. Physics Letters B, 286(3), 239-246. 93 [41] Janson, S. (2012). Simply generated trees, conditioned GaltonWatson trees, random allocations and condensation. Probab. Surv, 9, 103-252. 107 ¨ (2015). Scaling limits of random planar maps [42] Janson, S., & Stef´ansson, S. O. with a unique large face. The Annals of Probability, 43(3), 1045-1081. 14 [43] Krikun, M. (2005). Local structure of random quadrangulations. arXiv preprint math/0512304. 15 [44] Le Gall, J. F. (1999). Spatial branching processes, random snakes, and partial differential equations. Springer Science & Business Media. 4 [45] Le Gall, J. F. (2006). A conditional limit theorem for tree-indexed random walk. Stochastic processes and their applications, 116(4), 539-567. 6 [46] Le Gall, J. F. (2007). The topological structure of scaling limits of large planar maps. Inventiones mathematicae, 169(3), 621-670. 4, 6, 7, 11

158

[47] Le Gall, J. F. (2010). Geodesics in large planar maps and in the Brownian map. Acta mathematica, 205(2), 287-360. 7, 13 [48] Le Gall, J. F. (2013). Uniqueness and universality of the Brownian map. The Annals of Probability, 41(4), 2880-2960. 4, 7, 8, 10, 11, 12, 13, 14, 20, 29, 32, 34, 38, 41, 43, 51, 85, 93, 137 [49] Le Gall, J. F. (2014). Random geometry on the sphere. arXiv preprint arXiv:1403.7943. 4 [50] Le Gall, J. F., & Miermont, G. (2012). Scaling limits of random trees and planar maps. Probability and statistical physics in two and more dimensions, 15, 155-211. 4 [51] Le Gall, J. F. & Weill, M. (2006). Conditioned Brownian trees. Ann. IH Poincar´e-PR, 42, 455-489. 6, 11, 130 [52] Marckert, J. F., & Miermont, G. (2007). Invariance principles for random bipartite planar maps. The Annals of Probability, 35(5), 1642-1705. 6 [53] Marckert, J. F., & Mokkadem, A. (2006). Limit of normalized quadrangulations: the Brownian map. The Annals of Probability, 34(6), 2144-2202. 5 [54] McDiarmid, C. (1998). Concentration. In Probabilistic methods for algorithmic discrete mathematics (pp. 195-248). Springer Berlin Heidelberg. 73, 125 [55] M´enard, L. (2010). The two uniform infinite quadrangulations of the plane have the same law. In Annales de l’institut Henri Poincar´e (B) (Vol. 46, No. 1, pp. 190-208). 15 [56] Miermont, G. (2006). An invariance principle for random planar maps. DMTCS Proceedings, (1). 6 [57] Miermont, G. (2007). Invariance principles for spatial multitype Galton-Watson trees. Ann. Inst. H. Poincar´e Probab. Statist. 6 [58] Miermont, G. (2009). Tessellations of random maps of arbitrary genus. In An´ nales Scientifiques de l’Ecole Normale Sup´erieure (Vol. 42, No. 5, pp. 725-781). 52

159

[59] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta mathematica, 210(2), 319-401. 4, 8, 20, 38, 41, 43, 51, 85, 93, 137 [60] Miermont, G., & Weill, M. (2008). Radius and profile of random planar maps with faces of arbitrary degrees. Electron. J. Probab, 13(4), 79-106. 6 [61] Mohar, B., & Thomassen, C. (2001). Graphs on Surfaces (Vol. 10). JHU Press. 26 [62] Schaeffer, G. Conjugaison d’arbres et cartes combinatoires al´eatoires. PhD thesis, Universit´e Bordeaux I, 1998. 5, 15, 29 [63] Schramm, O. (2011). Conformally invariant scaling limits: an overview and a collection of problems. In Selected Works of Oded Schramm (pp. 1161-1191). Springer New York. 6 [64] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423-439. 73 [65] Tutte, W. T. (1963). A census of planar maps. Canad. J. Math, 15(2), 249-271. 20, 43, 58, 59, 62 [66] Walsh, T. R., & Lehman, A. B. (1975). Counting rooted maps by genus III: nonseparable maps. Journal of Combinatorial Theory, Series B, 18(3), 222-259. 62 [67] Weill, M. (2007). Asymptotics for rooted bipartite planar maps and scaling limits of two-type spatial trees. Electron. J. Probab, 12, 862-925. 6 [68] Wen, Y. (2015). The Brownian plane with minimal neck baby universe. arXiv preprint arXiv:1511.01028. x, 2, 22