Nov 3, 2010 - Abstract. Random Overlap Structures (ROSt's) are random elements on the space of probability measures on the unit ball of a Hilbert space,.
arXiv:1011.1823v1 [math.PR] 8 Nov 2010
Random Overlap Structures: Properties and Applications to Spin Glasses Louis-Pierre Arguin∗ and Sourav Chatterjee† Courant Institute of Mathematical Sciences New York University, New York, 10012, USA. November 3, 2010
Abstract Random Overlap Structures (ROSt’s) are random elements on the space of probability measures on the unit ball of a Hilbert space, where two measures are identified if they differ by an isometry. In spin glasses, they arise as natural limits of Gibbs measures under the appropriate algebra of functions. We prove that the so called ‘cavity mapping’ on the space of ROSt’s is continuous, leading to a proof of the stochastic stability conjecture for the limiting Gibbs measures of a large class of spin glass models. Similar arguments yield the proofs of a number of other properties of ROSt’s that may be useful in future attempts at proving the ultrametricity conjecture. Lastly, assuming that the ultrametricity conjecture holds, the setup yields a constructive proof of the Parisi formula for the free energy of the Sherrington-Kirkpatrick model by making rigorous a heuristic of Aizenman, Sims and Starr. ∗
L.-P. Arguin is supported by the NSF grant DMS-0604869 and partially by the Hausdorff Center for Mathematics, Bonn. † Sourav Chatterjee’s research was partially supported by NSF grants DMS-0707054 and DMS-1005312, and a Sloan Research Fellowship
1
1
Introduction
This paper develops some connections between random probability measures on Hilbert spaces and the Gibbs measures of spin glasses with Gaussian disorder. Consider for N ∈ N random processes of the form HN := (HN (σ), σ ∈ {−1, +1}N ) , where HN (σ) is a centered Gaussian variable. Denote the law of HN by P and the corresponding expectation by E. Suppose that the covariances between variables are of the form E HN (σ)HN (σ ′ ) = Nr(σ, σ ′ ) for some positive definite symmetric form r on {−1, +1}N with r(σ, σ) = 1 for all σ. We say that HN is the Hamiltonian of a Gaussian spin glass. An important example is the Sherrington-Kirkpatrick (SK) model for which r(σ, σ ′ ) =
N 1 X σi σi′ N i=1
!2
.
The definition also includes the Edwards-Anderson (EA) model. This model can be defined for example in a box of Zd with N vertices and edge set EN . The form for the EA model is then X 1 r(σ, σ ′ ) = σi σj σi′ σj′ . |EN | {i,j}∈EN
The Gibbs measure corresponding to the Hamiltonian HN at inverse temperature β is defined by Gβ,N (σ) =
exp βHN (σ) ZN (β)
(1.1)
P where ZN (β) = σ exp βHN (σ). A fundamental problem in the theory of spin glasses is to describe the limits of Gβ,N as N tends to infinity. This information is useful in particular to understand the extreme value statistics of the Gaussian process HN . In the past ten years, the rigorous study of spin glasses has made important progresses in the understanding of the models in the limit N → ∞. One of the major achievements was the proof of the 2
Parisi formula by Guerra and Talagrand [12, 27] for the limiting free energy (that is, limN →∞ N1 E log ZN (β)) of the SK model. The formula suggests that the limiting Gibbs measure Gβ of the SK model (in a suitable sense) has a support that is hierarchical or ultrametric, that is with P -probability one n o Gβ⊗3 (σ, σ ′ , σ ′′ ) : r(σ, σ ′ ) ≥ min {r(σ, σ ′′ ), r(σ ′, σ ′′ )} = 1 .
In fact, it is expected that the ultrametricity of the Gibbs measure is universal to a certain extent within spin glasses. This hypothesis is at the core of the description of spin glasses developed by physicists, and often referred to as Parisi theory [16]. In spite of important advances in the understanding of the structure of the Gibbs measure [20, 5], the problem of rigorously characterizing the limiting Gibbs measures of spin glasses remains open. This paper takes the approach introduced by Aizenman, Sims and Starr [3] and extended in [7] to study the limiting Gibbs measures of spin glasses. The idea goes as follows. Given an infinite-dimensional separable Hilbert space H with inner product “·”, there exists an embedding of the hypercube {−1, 1}N in H that preserves the form r, that is, if σ 7→ v(σ) ∈ H then r(σ, σ ′ ) = v(σ) · v(σ ′ ) . For example, for the SK model with N sites, the Hilbert space H can be taken to be the L2 space of some abstract probability space (Ω, F , P), and the map v defined as N 1 X v(σ) = gij σi σj , N i,j=1
where gij : Ω → R are independent standard Gaussian random variables under P. Under such an embedding, the Gibbs measure is naturally sent to the random probability measure on H assigning weight Gβ,N (σ) to the corresponding vector v(σ). Of course, there is more than one such isometric embedding. Since the products v · v ′ determine the vectors up to a choice of basis for H, it is necessary to identify the probability measures on H that differ by an isometry for the image measure of Gβ,N to be well-defined. It turns out that the proper way to study such measures is through the notion of a Random Overlap Structure or ROSt. A ROSt is a random element in the space of probability measures on the unit ball of an infinite-dimensional separable 3
Hilbert space with the identification that they are the same if they differ by an isometry. The space of ROSt’s is compact in the appropriate topology (see Section 1.1). Thus one way to study the limiting Gibbs measures of spin glasses is to describe the limit points of the sequence (Gβ,N ) in the space of ROSt’s. (Another promising approach is through the Ghirlanda-Guerra (GG) identities [13, 20, 28]. Although the GG identity approach is not the focus of this article, it seems to be closely connected to the ROSt approach at some level.) The main result of this paper is to establish that the limiting Gibbs measures (in the ROSt sense) of any Gaussian spin glass as defined above is stochastically stable. What this means is that the ROSt’s for Gaussian spin glasses are distributionally invariant under a built-in stochastic mapping that we call the cavity mapping. (The mapping is well-known, but the nomenclature is ours. Precise definition is given below.) The transformation derives its name and structure from the cavity method introduced in [17] (see also [9] for the definition of the mapping for Ruelle Probability Cascades). It has the features of a basic stochastic object akin to the mappings studied by Kahane in the context of multiplicative chaos [15]. Similar mappings have been considered earlier in the setting of competing particle systems (see e.g. [25, 5]). It was suggested by the work of several authors that the Gibbs measures of spin glasses must be stochastically stable [2, 3, 25, 5]. Such a statement, however, can be made precise only in the infinite-dimensional setting. One missing ingredient to make the assertion rigorous at the level of ROSt’s was a proof the continuity of the cavity mapping. This is done in Theorem 1.4. Combining this with some further ingredients, it follows that the limiting Gibbs measure of a Gaussian spin glass is stochastically stable at any β where the free energy is differentiable (by convexity, this holds for almost all β). The characterization of stochastically stable ROSt’s is a challenging problem. In view of the main result of this paper, it is also a rewarding one since it provides a way to establish universal properties of the Gibbs measures of spin glasses, in particular the alleged ultrametricity. We formulate below a precise conjecture, which may be called the Ultrametricity Conjecture for ROSt’s: a ROSt that is stochastically stable (in a strong sense to be defined) must have ultrametric support. Some partial results towards the resolution of this conjecture are given in Section 4. The most important one might be Theorem 1.7 which states that the support of a stochastically stable ROSt 4
is either one vector or lies in an infinite-dimensional subspace almost surely. The framework of ROSt’s has other interesting consequences for spin glasses. In particular, assuming that the ultrametricity conjecture for ROSt’s is true, we give a constructive proof of the Parisi formula for the free energy of the Sherrington-Kirkpatrick model. This makes rigorous the approach suggested by Aizenman, Sims and Starr [3] to prove the Parisi formula. We remark that results similar to ours have been proved in parallel by Panchenko [23]. The work focuses on mean-field spin glass models where the law of the Hamiltonian is invariant under permutation of spins, such as the SK model. In this case, a representation theorem for exchangeable arrays (the Aldous-Hoover theorem) can be used. In the ROSt setting, we appeal to the Dovbysh-Sudakov theorem (Theorem 1.2 below). A continuity result is proved and a representation for the Parisi formula, similar to the one obtained here for ROSt’s, is found The paper is organized as follows. The framework of ROSt’s as well as the precise statements of the main results are given in Section 1.1. The applications of the framework to spin glasses are presented in Section 1.2. The continuity of the cavity mapping is proved in Section 2. Section 3 derives the Parisi formula from the Ultrametricity Conjecture. Finally, some interesting properties of stochastically stable ROSt’s are proved in Section 4.
1.1
Random Overlap Structures and Probability Measure on Hilbert Spaces
This section begins with the definition of random overlap structures using exchangeable covariance matrices. The definition is then explained in terms of probability measures on Hilbert spaces using the representation theorem of Dovbysh and Sudakov [11]. This setup was previously described in [7]. The section ends with the definition of the cavity mapping and the statement of several related results. Consider the space of positive semi-definite symmetric N × N matrices, or covariance matrices, with 1 on the diagonal. This is a compact separable metric space when considered as a closed subset of [−1, 1]N×N equipped with the product topology. The Borel probability measures on this space form also a compact, separable and metric space when equipped with the weak-∗ topology generated by the continuous functions on the entries. An element P of this space is the law of some random covariance matrix Q = {qij }. The 5
continuous functionals in the topology considered can be approximated by linear combinations of monomials of the form Y k P 7→ E qijij , (1.2) 1≤i 0 be a parameter, P be the law of a ROSt with sampling measure µ, and l be a cavity field independent of P, we take: µ 7→ Φl µ :=
eλl(v)−
λ2 2
kvk2
2 λl(v)− λ2
µ(e
µ(dv) kvk2
.
(1.4)
)
The image ROSt is constructed from the sampling measure Φl µ. Its law is λ2 2 induced by P × Pl . We stress that the factor e 2 kvk is the expectation of eλl(v) , and its presence turns out to be necessary for the continuity of the mapping. The cavity mapping can also be defined in terms of the continuous functions (1.3): # " 2 2 λl(vs )− λ2 kvs k2 λl(v1 )− λ2 kv1 k2 ⊗s ...e ) µ (F (v)e . E[µ⊗s (F (v))] 7→ EEl λ2 λ2 1 1 2 s s 2 µ⊗s (eλl(v )− 2 kv k ...eλl(v )− 2 kv k ) It will be a consequence of Theorem 1.4 that the cavity mapping is welldefined , that is, the image ROSt is the same if we apply (1.4) to two random elements of M(B) that are sampling measures of the same ROSt. We remark that this mapping can be generalized in at least two ways. First, for any function ψ ∈ C 1 (R) √ with bounded derivative, we can take ψ(l(v)+z 1−kvk2 ) the change of density Ez [e ], where z is a standard Gaussian with expectation Ez . Equation (1.4) corresponds to the choice ψ(x) = λx. Second, one could consider the cavity field lc with El [lc (v)lc (v ′ )] = c(v · v ′ ) for some function c satisfying c(0) = 0 c(1) = 1 (v, v ′ ) 7→ c(v · v ′ ) is positive definite .
(1.5)
We will often take lp which corresponds to the choice c(v · v ′ ) = (v · v ′ )p for p ∈ N. The first result of this paper, from which all others essentially stem, is proved in Section 2. Theorem 1.4. The mapping (1.4) from the space of ROSt’s to itself is continuous. 8
The result also holds for the generalizations discussed above. Our focus is on ROSt’s that are invariant under the cavity mapping. Definition 1.5. A ROSt is called stochastically stable if its law is invariant under the cavity mapping (1.4) for any λ > 0. By Proposition 1.3, to say a ROSt is stochastically stable is equivalent to say that its sampling measure µ has the same law as Φl µ, up to an isometry of B. As we shall explain in the next section, continuity provides the missing link between the stochastic stability as defined here and stochastic stability of the Gibbs measure of spin glasses as studied in [2, 10, 28]. This link provides a statement of the ultrametricity conjecture on the characterization of stochastically stable ROSt’s. Conjecture 1 (Ultrametricity Conjecture). If a ROSt of law P is stochastically stable under the cavity fields lp (defined below (1.5)) for p = 1 and infinitely many p, then the support of its sampling measure µ is ultrametric P-a.s., that is: v · v ′ ≥ min {v · v ′′ , v ′ · v ′′ } for µ-almost all v, v ′, v ′′ . Moreover, the law of the sampling measure is a convex combination of Ruelle Probability Cascades. Ruelle Probability Cascades form a compact subset of ROSt’s parametrized by the right-continuous, increasing functions on [0, 1] with values 0 at 0 and 1 at 1 (we refer to [7] for a proof of the compactness). They have ultrametric support and they are known to be stochastically stable for any cavity field lp , p ∈ N. We shall not need their precise definition here. The reader is referred to [24, 9, 5] for more details. It is very likely that the hypotheses of the conjecture can be relaxed. However, it does not hold if stochastic stability for a single cavity field is assumed. A simple counterexample is given at the end of Section 4. The conjecture was proved in the case where the support of µ⊗2 is finite in [5] (see also [18] for an extension of that proof). In this case, µ is supported on countable vectors and the conjecture can be formulated in the language of competing particle systems. A similar statement of the ultrametricity conjecture exists where the assumption of stability is replaced with the extended Ghirlanda-Guerra identities:
9
Definition 1.6. A ROSt with sampling measure µ is said to satisfy the Ghirlanda-Guerra (GG) Identities if for any s ∈ N and any continuous function F on s replicas we have E[µ⊗s+1 (v 1 · v s+1 F (v))]
s
1 1X = E[µ⊗2 (v 1 · v 2 )] E[µ⊗s (F (v))] + E[µ⊗s (v 1 · v l F (v))] . s s l=2
It is said to satisfy the extended Ghirlanda-Guerra identities if the above holds when the products v i · v j in the expectations is replaced by (v i · v j )p for any p ∈ N. A version of the ultrametricity conjecture assuming finiteness of the support of the overlap distribution and the extended Ghirlanda-Guerra identities was proved by Panchenko in the case where µ⊗2 has finite support [20]. The assumptions are in contrast with the ones used in the proof of the conjecture using stochastic stability in [5] where stability for infinitely many p’s in N, but not all p, is needed. The continuity of the mapping has interesting consequences that we discuss in Section 4. For one, it provides an ergodic decomposition of stochastically stable ROSt’s. The Ultrametricity Conjecture can then be restated by saying that the extremes of the considered set of stochastically stable ROSt’s are exactly the RPC’s. The decomposition is useful to prove properties of the support of the sampling measure. We stress that stochastic stability for a single cavity field is necessary here. Theorem 1.7. The sampling measure of a stochastically stable ROSt is supported on a single vector or on an infinite-dimensional subset of B. If stability under more fields is assumed, we can prove more. Theorem 1.8. If a ROSt is stochastically stable under the cavity field lp for an infinite number of p ∈ N, then its sampling measure is supported almost surely on a sphere. This last property does not hold in general if stability is assumed for a single cavity field. A counter-example is given in Section 4. Properties of the sampling measure for ROSt’s satisfying the extended Ghirlanda-Guerra identities have been obtained in [20]. The apparent similarities of the properties of the ROSt’s satisfying the Ghirlanda-Guerra 10
identities and stochastic stability motivate the following question: are the Ghirlanda-Guerra identities and stochastic stability two representations of the same property? Since the set of stochastic stable ROSt’s is closed under convex combinations and not the set of ROSt’s satisfying GG, we conjecture the following: Conjecture 2. The ROSt’s satisfying the Ghirlanda-Guerra identities are the extremes of the convex set of stochastically stable ROSt’s.
1.2
Application to Spin Glasses
Consider now a spin glass HN and its Gibbs measures Gβ,N at β as defined in the introduction. It is straightforward to construct a ROSt Qβ,N from Gβ,N by taking Qβ,N = { r(σ i , σ j ) }i,j∈N
where (σ i )i∈N are elements of {−1, +1}N sampled iid under Gβ,N . Here the form “r” is the one of the given spin glass model, but he same construction holds for any positive definite symmetric form on {−1, +1}N . Note that the randomness of Qβ,N comes from the sampling and the randomness of Gβ,N itself. The considerations of the previous sections ensures that the random matrix Qβ,N can be constructed from a random sampling measure µβ,N on B. In particular, for any function F on s replicas there is the identity ⊗s k l E Gβ,N (F (r(σ k , σ l ); 1 ≤ k < l ≤ s)) = E µ⊗s β,N (F (v · v ; 1 ≤ k < l ≤ s)) .
The sequence (Qβ,N ) in the space of ROSt’s has limit points, since the space is compact. These limits should express in some ways the Gibbs nature present at finite N. Two properties of the limit points seem to be of importance: the Ghirlanda-Guerra identities as presented earlier and the stochastic stability for the sequence of sampling measure (µβ,N ). Stochastic stability for the sequence defined below does not directly translate into stochastic stability as presented earlier of the limit ROSt. As we shall see, Theorem 1.4 fills this gap. Stochastic stability was first introduced in [2] and originally defined as follows. Let (Gβ,N ) be a sequence of Gibbs measures at inverse temperature β. Consider similarly as before the Gaussian field (l(σ) : σ ∈ {−1, +1}N ) independent of HN , whose law we again denote by El , with covariance 11
El [l(σ)l(σ ′ )] = σ · σ ′ . For some λ > 0, we consider the mapping (1.4). Since here kσk = 1 for any vector in the support, it reduces to Gβ,N (σ) 7→
Gβ,N (σ)eλl(σ) . Gβ,N (eλl(σ) )
(1.6)
It is readily checked by property pof Gaussians that the image in (1.6) is the Gibbs measure at temperature β 2 + λ2 /N. The original idea of [2] is that a continuous dependence on β of the Gibbs measure is equivalent in the limit N → ∞ to stability of the measure under the mapping (1.6). Definition 1.9. A sequence of Gibbs measures (GN ) is said to be stochastically stable if for any λ > 0, s ∈ N and for any continuous function F on s replicas, " # 1 s ⊗s GN (F (σ)eλl(σ ) ...eλl(σ ) ) ⊗s lim EEl − EGN (F (σ)) = 0 , 1) s) ⊗s λl(σ λl(σ N →∞ GN (e ...e ) where F (σ) stands for F (r(σ l , σ k ); 1 ≤ k < l ≤ s).
It was shown in [10] that stochastic stability in the sense of Definition 1.9 holds for almost all β. An improvement was made in [28] where it is proved that for any β and sequence of Gibbs measures (Gβ,N ) there exists a sequence βN → β such that the sequence (GβN ,N ) is stochastically stable. We show here that stochastic stability holds at any β where limN →∞ N1 E log ZN (β) is differentiable. This is an analogue of an elegant result of Panchenko [21] which proves the validity of the Ghirlanda-Guerra identities under the same hypothesis. Proposition 1.10. If limN →∞ N1 E log ZN (β) exists and is differentiable at β > 0, then the sequence of Gibbs measures (Gβ,N ) is stochastically stable in the sense of Definition 1.9. Proof. Let F be a continuous function on s replicas. We must have |F | ≤ C for some C > 0. Straightforward differentiation yields h i 1 ⊗s ⊗s ∂β EGβ,N (F (σ)) = s EGβ,N HN (σ ) − Gβ,N HN (σ) F (σ) . Hence the absolute value of the derivative is bounded by sC EGβ,N HN (σ 1 ) − Gβ,N HN (σ) . 12
Define β(λ) :=
p β 2 + λ2 /N. By integration, we have the bound
⊗s ⊗s F (σ) EGβ(λ),N F (σ) − EGβ,N Z β(λ) ≤ sC EGβ ′ ,N |HN (σ) − Gβ ′ ,N (HN (σ))| dβ ′ . (1.7) β
It remains to show that the integral goes to zero under the assumption. We have by Cauchy-Schwarz inequality Z
β(λ)
EGβ ′ ,N |HN (σ) − Gβ ′ ,N (HN (σ))| dβ ′
β
≤N
1/2
β(λ) − β
1/2
Z
β(λ) β
1 EGβ ′ ,N |HN (σ) − Gβ ′ ,N (HN (σ))|2 dβ ′ N
!1/2
.
2
λ Since β(λ) − β = 2βN + O( N12 ), stochastic stability in the sense of Definition 1.9 would follow by equation (1.7) if the term in the parentheses divided by N goes to zero as N → ∞. Define fN (β) := N1 E log ZN (β). fN is famously convex. It is easily checked by differentiation that the integral to bound is exactly fN′ (β(λ)) − fN′ (β).
Since β(λ) → β, the above goes to zero by a simple result of convexity (see Lemma 3.5) as well as the assumptions on the existence of the limit and the derivative at β. A direct consequence of Theorem 1.4 is that stochastic stability as a property of the sequence becomes a property of the limit points. Corollary 1.11. If limN →∞ N1 E log ZN (β) exists and is differentiable at β, then the limit points in the space of ROSt’s of the sequence (Qβ,N ) are stochastically stable in the sense of Definition 1.5. Another notable application of the continuity of the cavity mapping is a proof that the Parisi functionals, entering in the Parisi formula of the SK model, are continuous as functionals on ROSt’s. This is proved in Section 2.2 and can be seen as a generalization of a theorem of Guerra that proved the continuity within the subset of Ruelle Probability Cascades [12, 4]. Assuming that the Ultrametricity Conjecture holds, this fact, together with Guerra’s 13
bound and Corollary 1.11, provides a proof of the Parisi formula for the SK model (and more generally, for mixed p-spin models with even p’s). This is done in Section 3.
2 2.1
The Cavity Mapping Continuity
In this section, we prove the continuity of the mapping (1.4). Let ψ be a function in C 1 (R) with bounded derivative, |ψ ′ | ≤ C for some C > 0. Assume without loss of generality that ψ(0) = 0. Let z a standard Gaussian variable with law Pz and expectation Ez . Note that, under these assumptions, |ψ(z)| ≤ C|z|, and in particular Ez [emψ(z) ] < ∞ for any m ∈ R. Define √ 2 (2.1) G(v, z, l) := eψ(l(v)+z 1−kvk ) . In the cases ψ(x) = x and ψ(x) = log cosh x, we have the simplification 1
2)
Ez G(v, z, l) = eψ(l(v)) e 2 (1−kvk
.
When dealing with s replicas, that is v = (v 1 , ..., v s ) ∈ Bs and z = (z 1 , ..., z s ) where z has law Pz⊗s , write G(v, z, l) :=
s Y
G(v i , z i , l) .
(2.2)
i=1
Consider the following generalization of the mapping (1.4): µ 7→ Φl µ :=
µ(dv) Ez G(v, z, l) , µ(Ez G(v, z, l))
(2.3)
where l(v) is the cavity field with covariance v·v ′ . The proof of the continuity is the same when the covariance of the field is c(v · v ′), for a suitable function c, if kvk2 in the definition of G is replaced by c(kvk2 ). We restrict ourselves to the case where c is the identity to simplify notation. Consider a function F = F (v, z, l) that depends measurably on s replicas, the Gaussian vector z and the cavity field. The main ingredient of the proof of continuity is an approximation in the spirit of the weak law of large numbers. 14
Lemma 2.1. Let µ ∈ M(B). Let F (v, z, l) be a function such that v 7→ El Ez⊗s [F (v, z, l)2 ] is a continuous function on s replicas. Let (v r , z r ), r = 1, ..., n, be n independent copies of (v, z) sampled under (µ × Pz )⊗s . Then El (µ × Ez )⊗ns
! n 2 C 1X F (v r , z r , l) − (µ × Ez )⊗s F (v, z, l) ≤ n r=1 n
where C is a positive constant that depends on F but not on µ. Proof. Using the fact that under (µ × Ez )⊗ns , the variables F (v r , z r , l), r = 1, ..., n are i.i.d., we have ! n 2 X 1 F (v r , z r , l) − (µ × Ez )⊗s F (v, z, l) (µ × Ez )⊗ns n r=1 1 2 1 = (µ × Ez )⊗s F (v, z, l)2 − (µ × Ez )⊗s F (v, z) n n 1 ≤ (µ × Ez )⊗s F (v, z, l)2 . n Now integrate over z and over the l’s using Fubini’s theorem. By assumption, v 7→ El Ez⊗s [F (v, z, l)2 ] is a bounded function on Bs . The conclusion follows from this.
It will be useful to impose stronger conditions pon F (v, z, l), namely that i i F (v, z, l) is a function of the variables l(v ) + z 1 − kv i k2 , i = 1, ..., s: p (2.4) F (v, z, l) = F (l(v i ) + z i 1 − kv i k2 , i = 1, ..., s) ;
and that for any m ∈ R,
v 7→ El Ez⊗s [F (v, z, l)m ] is a continuous function on s replicas.
(2.5)
Lemma 2.2. The function G(v, z, l) satisfies (2.4) and (2.5). i Proof. p The dependence in G is uniquely on the Gaussian variables l(v ) + i 2 z 1 − kvk , i = 1, ..., s. These variables have variance 1 and covariance v i · v j , i 6= j. Therefore the function v = (v 1 , ..., v s ) 7→ El Ez⊗s [G(v, z, l)m ] depends only on the inner products between distinct replicas v 1 ,...,v s . By writing down the Gaussian integral El Ez⊗s explicitly, we see that to prove
15
it is continuous on Bs it suffices to show that, for any m ∈ R, the function v 7→ El Ez⊗s [G(v, z, l)m ] is bounded uniformly in v. By H¨older’s inequality, # " s s Y Y 1/s ⊗s i i m ≤ El Ez i [G(v i , z i , l)ms ] = Ez [esmψ(z) ] < ∞, El Ez G(v , z , l) i=1
i=1
which shows the desired uniform bound.
Theorem 1.4 is a straightforward consequence of the following result with F (v, z, l) = F (v). The idea of the proof is to linearize functions involving the measure µ × Ez using the empirical approximation of Lemma 2.1. Theorem 2.3. Let F (v, z, l) be a function satisfying (2.4) and (2.5) and let G(v, z, l) be as in (2.2). Then the mapping (µ × Ez )⊗s F (v, z, l)G(v, z, l) (2.6) P 7→ EEl (µ × Ez )⊗s G(v, z, l) is a continuous functional on the space of ROSt’s.
Proof. We rearrange the image of the mapping, (µ × Ez )⊗s F (v, z, l)G(v, z, l) = EEl ⊗s G(v, z, l) (µ × Ez ) 1 Pn r r r r r=1 F (v , z , l)G(v , z , l) + dµ (F G) ⊗ns n Pn EEl (µ × Ez ) 1 r r r=1 G(v , z , l) + dµ (G) n (2.7) where n 1X ⊗s F (vr , z r , l)G(vr , z r , l) dµ (F G) := (µ × Ez ) F (v, z, l)G(v, z, l) − n r=1 n 1X dµ (G) := (µ × Ez )⊗s G(v, z, l) − G(v r , z r , l) . n r=1 We introduce the empirical average of f as a function on the n copies (v r , z r )nr=1 and the cavity field Pn 1 F (v r , z r , l)G(v r , z r , l) P F˜ ((v r , z r )r ; l) := n r=1 . (2.8) n 1 r r r=1 G(v , z , l) n 16
In this notation, elementary manipulations of (2.7) give (µ × Ez )⊗s F (v, z, l)G(v, z, l) − EEl (µ × Ez )⊗ns F˜ ((v r , z r )r ; l) EEl (µ × Ez )⊗s G(v, z, l) (µ × Ez )⊗ns dµ (F G) − F˜ ((v r , z r )r ; l) dµ (G) . = EEl ⊗s (µ × Ez ) G(v, z, l) (2.9) We now bound the absolute value of the difference appearing in the lefthand side of (2.9) uniformly in √the possible laws P. We write for simplicity √ 2 . We use H¨older’s inequality followed by γ = 2 and γ¯ = (1 − γ1 )−1 = √2−1 Jensen’s inequality and Fubini’s theorem to get the following upper bound of the right-hand side: n
Eµ
⊗ns
El Ez⊗ns |dµ (F G)
o1/γ r r γ ˜ − F ((v , z )r ; l) dµ (G)| o1/¯γ n × Eµ⊗ns El Ez⊗ns [G(v, z, l)−¯γ ] .
The second term is bounded by a constant uniform in the laws P by Lemma 2.2. (essentially, the term with negative power −¯ γ is bounded because the Laplace functional of a Gaussian variable is well-defined everywhere.) As for the first term, by the triangle inequality, it is smaller than n
Eµ
⊗ns
El Ez⊗ns |dµ (F G)|γ
o1/γ
o1/γ n ⊗ns ˜ r r γ ⊗ns El Ez |F ((v , z )r ; l) dµ (G)| + Eµ .
The first term is bounded by the same√expression with γ replaced by 2. By Lemma 2.1, it is thus bounded by C/ n uniformly in P. The second term is bounded similarly after an application of H¨older’s inequality with γ. We note that in this case, the resulting term Eµ⊗ns El Ez⊗ns |F˜ ((v r , z r )r ; l)|γ¯γ is bounded uniformly in the P’s by (2.5), Lemma 2.2 and the use of Jensen’s inequality in (2.8). Putting all this together, we can write from (2.9) for a
17
possibly different C, not depending on P, ⊗s F (v, z, l)G(v, z, l) (µ × E ) z ⊗ns r r EEl − Eµ El Ez [F˜ ((v , z )r ; l)] (µ × Ez )⊗s G(v, z, l) C ≤√ . n
(2.10)
The above approximation is useful because it is uniform in the laws P, and it linearizes the dependence on µ (allowing the use of Fubini’s theorem to exchange the integration El and µ). If E and E′ are two ROSt’s with sampling measures µ and µ′ , we have by (2.10) ⊗s (µ × E ) F (v, z, l)G(v, z, l) z EEl ⊗s (µ × Ez ) G(v, z, l) (µ′ × Ez )⊗s F (v, z, l)G(v, z, l) −E′ El ′ ⊗s (µ × Ez ) G(v, z, l) ⊗ns ⊗ns ˜ r r ⊗ns ˜ r r ′ ′⊗ns El Ez [F ((v , z )r ; l)] El Ez [F ((v , z )r ; l)] − E µ ≤ Eµ 2C +√ . n
Since C is uniform in the choice of P, the continuity will follow if (v r )nr=1 7→ El Ez⊗ns [F˜ ((v r , z r )r ; l)] is a continuous function on ns replicas. We have that 1 Pn r r r r r=1 F (v , z , l)G(v , z , l) ⊗ns ˜ r r ⊗ns n Pn . El Ez [F ((v , z )r ; l)] = El Ez 1 r , z r , l) G(v r=1 n
But p the integration is on the ns Gaussian variables ′ of the form l(v) + z 1 − kvk2 that have variance 1 and covariance v · v . Therefore the expectation is a function on ns replicas, since it depends only on the inner products between distinct replicas and not on their norm. It is readily seen 18
from writing the Gaussian integral that the continuity will follow if the expectation is bounded on Bns . We have by H¨older’s inequality and Jensen’s inequality 1/3 El Ez⊗ns [F˜ ((v r , z r )r ; l)] ≤{El Ez⊗s F (v, z, l)3 }1/3 El Ez⊗s G(v, z, l)3 × {El Ez⊗s G(v, z, l)−3 }1/3 .
All terms are bounded as function on Bns by (2.5) and Lemma 2.2.
2.2
The Parisi Functionals
Theorem 2.3 that proves the continuity of the mapping also provides continuity for an important class of functionals on ROSt’s, the so-called Parisi functionals. They enter in particular in the Parisi formula for the free energy of the SK model. Let ψ ∈ C 1 (R) with bounded derivative and with ψ(0) = 0. Consider a cavity field l with covariance El [l(v)l(v ′ )] = c(v · v ′ ) for some function c with c(0) = 0 and c(1) = 1, that preserves positive definiteness. Define the Parisi functional for a ROSt P and parameter λ ≥ 0 as i h √ 2 (2.11) P(λ, P) := E El log (µ × Ez ) eψ(λl(v)+λz 1−El [l(v) ])
where z is a standard Gaussian variable with expectation Ez . We drop the dependence on the c and on ψ in the notation for simplicity. The Parisi functionals evaluated on Ruelle Probability Cascade in fact equals f (0, 0), where f (q, y) is the solution of a p.d.e. with final condition f (1, y) = ψ(λy), see e.g. [3, 4] for details. Continuity of the Parisi functional on this subset of the space of ROSt’s has been established by Guerra [12]. It turns out that properties of the Parisi functionals that are needed in the proof of the Parisi formula can be established for general ROSt’s. Proposition 2.4. Let P(λ, P) be as in (2.11). Then, for any ψ ∈ C 1 (R) with bounded derivative, 1. P 7→ P(λ, P) is a continuous functional on ROSt’s for every λ ≥ 0. 2. λ 7→ P(λ, P) is a continuous functional on [0, ∞) for any ROSt of law P. In fact, for any δ > 0 and λ0 ≥ 0, if |λ − λ0 | < δ, then max |P(λ, P) − P(λ0 , P)| ≤ K|λ − λ0 | P ROSt
where K > 0 depends on λ0 and δ. 19
Proof. We prove the first claim. We show that P(λ, P) is differentiable in λ and that the derivative is a continuous function on ROSt’s. The conclusion follows by integration. We write for simplicity p g(v, z) := l(v) + z 1 − c(kvk2 ) ,
omitting the dependence on l. Note that under El Ez , the g’s are all centered Gaussian variables of variance 1. Suppose first that the derivative can be taken inside the expectations E El and µ × Ez , then one would get (µ × Ez ) g(v, z)ψ ′ (λg(v, z))eψ(λg(v,z)) . ∂λ P(λ, P) = EEl (2.12) ψ(λg(v,z)) (µ × Ez ) e By Theorem 2.3, the derivative will be continuous as a functional on ROSt’s if g(v, z)ψ ′(λg(v, z)) satisfies (2.4). This is straightforward since ψ ′ is bounded. It remains to show (2.12). The derivative can be taken inside the expectation EEl if we show that for δ small enough 2 ψ(λg(v,z)) (µ × Ez ) e 1 log < ∞ . (2.13) sup EE l 2 ψ(λ g(v,z)) 0 |λ−λ0 | 1}. The expectation in (2.13) becomes EEl (log R(λ, λ0 ))2 ; R(λ, λ0) > 1 h i 2 + EEl log R(λ, λ0 )−1 ; R(λ, λ0)−1 > 1 .
20
Since log x ≤ x − 1 for x > 1, we can bound the above by 2 (µ × Ez ) eψ(λg(v,z)) − eψ(λ0 g(v,z)) + EEl (µ × Ez ) eψ(λg(v,z)) 2 (µ × Ez ) eψ(λg(v,z)) − eψ(λ0 g(v,z)) . (2.14) EEl (µ × Ez ) eψ(λ0 g(v,z))
Using successively Cauchy-Schwarz inequality followed by Jensen’s inequality on each term yields the upper bound
ψ(λg(v,z))
EEl (µ × Ez ) e
ψ(λ0 g(v,z))
−e
4 1/2
i1/2 h × EEl (µ × Ez ) e−4ψ(λg(v,z))
h i1/2 −4ψ(λ0 g(v,z)) + EEl (µ × Ez ) e (2.15)
The terms in the second bracket can be integrated by Fubini’s and since the fields g(v, z) all have variance 1 under El Ez , the dependence on E drops leaving 1/2 1/2 Ez e−4ψ(λz) + Ez e−4ψ(λ0 z) .
The fact that |ψ ′ | ≤ C and ψ(0) = 0 implies that |ψ(λz)| ≤ λC|z|. And since |λ − λ0 | < δ, |ψ(λz)| ≤ C(λ0 + δ)|z|. Hence after integration, the term in the parentheses is seen to be uniformly bounded by a term that depends only on λ0 and δ. It remains to bound the first term. After a use of Fubini’s theorem, since the fields g(v, z) have variance 1, it is simply n 4 o1/2 Ez eψ(λz) − eψ(λ0 z) . (2.16)
Taylor’s expansion around ψ(λ0 z) gives for a certain z¯ between λ0 z and λz, eψ(λz) − eψ(λ0 z) = (λ − λ0 )zψ ′ (¯ z )eψ(¯z ) .
(2.17)
The latter is bounded above by C|λ − λ0 | |z|eC|¯z | . The assumption on z¯ readily implies that |¯ z | ≤ |z|(λ0 + δ). Plugging all this back into (2.16), we 21
get a bound after integration of z which is (λ − λ0 )2 times a constant that depends only λ0 and δ . Since the term (λ − λ0 )2 cancels with the one in (2.13), it follows that the differential operator can be passed through EEl . To prove (2.12), it remains to show (µ × Ez ) g(v, z)ψ ′ (λg(v, z))eψ(λg(v,z)) ∂λ log(µ×Ez ) eψ(λg(v,z)) = . (2.18) (µ × Ez ) eψ(λg(v,z))
This is straightforward once it is established that ′ ψ(λg(v,z)) ψ(λg(v,z)) . = (µ × Ez ) g(v, z)ψ (λg(v, z))e ∂λ (µ × Ez ) e
To do so, it suffices to justify taking the derivative inside µ × Ez by showing 2 1 ψ(λg(v,z)) ψ(λ0 g(v,z)) sup (µ × E ) e − e 0 we can find M(δ) independently of N such that for M > M(δ) ZN +1 (β + ) N X +2 (βp − βp2 ) < δ . 0 ≤ E log ≤ 2 ZN +1 (β + ) M p>M
Fix such a δ and such a M. We write for short fN (β) for N1 E log ZN (β). By the mean-value theorem on RM , there exists β M with βp ≤ β¯p ≤ βp+ for p ≤ M and β¯p = βp for p > M such that X ZN +1 (β + M) = N(βp+ − βp )Dp fN (β M ) . E log ZN +1 (β) p≤M We know that fN (β) is a convex function of β. Thus, by Lemma 3.5 and the assumption on differentiability of f (β), X (p − 1)βp X lim lim Dp fN (β) N(βp+ − βp )Dp fN (β M ) = N →∞ N →∞ 2 p≤M p≤M =
X (p − 1)βp2 1 − EGβ⊗2 (v · v ′ )p , 2 p≤M
28
where we used (3.3) in the last equality as well as the continuity of the function EG ⊗2 (v · v ′ )p in the space of ROSt’s. Since δ was arbitrary, the theorem is proved. Theorem 3.4 becomes useful once properties of the limiting Gibbs measure are known. This is achieved by a direct application of Corollary 1.11. Lemma 3.6. Let β with βp > 0 for all p be such that Dp f (β) exists for p = 1 and all even p′ s. Let Gβ be any limit point of (Gβ,N ) in the space of ROSt’s. Then, for p = 1 and all even p’s, Gβ is stochastically stable for the cavity field lp of covariance El l(v)l(v ′ ) = (v · v ′ )p , For the proof of the Parisi formula, we shall need the bound proved by Guerra [12]. Theorem 3.7 (Guerra’s bound). For any RPC with law denoted by P, f (β) ≤ log 2 + P(kβk, P) −
X (p − 1)βp2 p
2
1 − Eµ⊗2 (v · v ′ )p .
The proof is based on an interpolation between a cascade with a finite number of levels and the the Gibbs measure Gβ,N . It uses explicitly the tree structure of the cascades. The proof of this bound can be also done using the approach of Aizenman, Sims and Starr [3]. In this case, the proof does not use the ultrametric structure of the cascade explicitly, but simply the stability of the cascades under the cavity mapping with function ψ(x) = log cosh x and ψ(x) = x. The reader is referred to [3, 4] for more details. Proof of Theorem 3.1 using Conjecture 1. By Conjecture 1, the only stochastically stable ROSt’s for infinitely many p’s are convex combinations of RPC’s. In particular, by Lemma 3.6, at any point of differentiability β with βp > 0 for all p, the limit points of (Gβ,N ) are convex combinations of RPC’s. Therefore, by Theorem 3.4 and Guerra’s bound, the Parisi formula is established at any point β of differentiability with βp > 0 for all p. (Note that the functionals that are minimized in the Parisi formula are linear functions in the law of the ROSt’s. In particular, the minimum can be restricted to RPC’s as opposed to convex combinations of RPC’s.) It remains to show that the Parisi formula holds at all β, including those β with βp = 0 for some p. Since f is continuous in the space of parameters satisfying (3.2) and 29
since the point of differentiability β are dense in that space by Lemma 3.3, it suffices to show that the functional ( ) X (p − 1)βp2 β 7→ min P(kβk, P) − 1 − Eµ⊗2 (v · v ′ )p (3.11) P RPC 2 p P is continuous under the norm kβk2 = p 2p βp2 . We write for convenience Λ(β, P) for the functional in the argument of the minimum. By Proposition 2.4, Λ(β, ·) is a continuous functional on ROSt’s. We denote by Pβ the RPC where the minimum of Λ(β, ·) is attained. Since we minimize Λ over ROSt’s, we have trivially Λ(β, Pβ )−Λ(β 0 , Pβ ) ≤ min Λ(β, P)− min Λ(β 0 , P) ≤ Λ(β, Pβ0 )−Λ(β 0 , Pβ0 ) . P RPC
P RPC
(3.12)
On the other hand, we have for some K > 0, max |Λ(β, P) − Λ(β 0 , P)| ≤ K kβk − kβ 0 k .
P ROSt
This is because, by the second part of Proposition (2.11), max |P(kβk, P) − P(kβ 0 k, P)| ≤ K kβk − kβ 0 k , P ROSt
and the same holds by inspection for the second functional entering in the definition of Λ. The continuity of (3.11) in the space of β satisfying (3.2) follows from (3.12) and the two above estimates.
4
Properties of Stochastically Stable ROSt’s
Some particular properties of stochastically stable ROSt’s were essential in the proof of the ultrametricity conjecture in [5] and [20], where strong assumptions on the ROSt were needed (e.g., finiteness of the support for the law of the entries). Here we show similar properties in a general settting which may be useful for the proof of the conjecture in the general case. A direct consequence of the continuity of the cavity mapping is the compactness of the set of stochastically stable ROSt’s. Corollary 4.1. The set of stochastically stable ROSt’s is convex and compact. So is the set of ROSt’s that are stochastically stable under the cavity fields lc with El [lc (v)lc (v)] = c(v · v ′ ) for an infinite (possibly uncountable) collection of functions c’s satisfying (1.5). 30
Proof. Convexity is clear. The compactness of the set of ROSt’s that are invariant under the mapping (1.4) for a given cavity field is a consequence of continuity of such mappings (Theorem 1.4). Compactness also holds for sets of ROSt’s that are invariant under an infinite number of mappings (e.g., for all parameters λ > 0) since in a compact space, an arbitrary intersection of compact sets are compact (the intersection being here on the parameter λ of the cavity mapping and the family of c’s). Corollary 4.1 yields a standard ergodic decomposition for a stochastically stable measure by Choquet’s theorem. We will write Ext(SS) for the set of extreme points of the compact set of stochastically stable ROSt’s. Choquet’s theorem implies that any stochastically stable ROSt can be written as the barycenter of some probability measure on Ext(SS). Definition 4.2. We say function h : M(B) → R is an invariant function of the cavity mapping Φl defined by equation (1.4) if: 1) it is Borel measurable; 2) for any isometry T of B, h(µ) = h(T −1 µ); 3) h(µ) = h(Φl µ) Pl -almost surely. A standard consequence of the Choquet decomposition is the following. Lemma 4.3. If P ∈ Ext(SS), then any invariant function h : M(B) → R, of the cavity mapping is constant P-almost surely. Proof. If h is not a constant, we have the non-trivial decomposition of P Z P= P( · | h(µ) = y) Ph−1 (dy) . (4.1) R
On the other hand, P( · |h(µ)) is a stochastically stable ROSt for almost all h(µ). Indeed, for any Borel measurable function g on M(B), invariant under isometry, E[g(µ)h(µ)] = EEl [g(Φl µ)h(Φl µ)] = EEl [g(Φl µ)h(µ)] , where the first equality follows by stochastic stability and the second, from the invariance of h. Thus (4.1) contradicts the assumption that P ∈ Ext(SS) and the claim follows. By inspection of the form of the cavity mapping (1.4), we expect that the support of a measure µ is not affected by the cavity mapping, since the mapping modifies only the weight of the vectors and not their relative position. We make this idea precise in the following lemma. 31
Lemma 4.4. Let µ ∈ M(B). Then Φl µ is equivalent to µ Pl -almost surely, that is: for any non-negative measurable function f on B, Z Z f (v) µ(dv) = 0 if and only if f (v)Φl µ(dv) = 0 Pl -a.s. B
B
Proof. First, we have that Z Z f (v)eλl(v) f (v)Φl µ(dv) = µ(dv) , λl(v) ) B B µ(e R hence the left-hand side is zero if and only if f (v)eλl(v) µ(dv) = 0. Now if B R = 0, then we must have that f (v) = 0 µ-a.e., which implies RB f (v) µ(dv) λl(v) f (v)e µ(dv) = 0. On the other hand, we note that eλl(v) B R > 0 P2l -a.s. and µ-a.e. This is because l(v) is finite a.e., since by Fubini, El B |l(v)| µ(dv) ≤ R 1. Therefore if B f (v)eλl(v) µ(dv) = 0, then f (v) = 0 µ-a.e. also.
Functions of the support typically turn out to be invariant functions of the mapping. This simple fact can be used to investigate the support of stochastically stable ROSt’s. Here we look at two examples: the dimension of the support of the measure and whether or not the support lies on a sphere. We remark that the parameter λ of the cavity mapping will play no role in the proofs, and that the results actually hold when there is invariance for a single value of the parameter. We drop it for the rest of the section for simplicity. We define the dimension of the support of the measure as the dimension of the smallest subspace that contains the support. One way to make this precise is as follows [1]. Let µ ∈ M(B). Define (weakly) the covariance operator Cˆµ on H as Z ′ ′′ ′ ′′ ˆ µ(dv) (v ′ · v)(v · v ′′ ) . for all v , v ∈ H, v · (Cµ v ) := B
It is easily checked that Cˆµ is a self-adjoint, trace-class linear operator on P H. (It is trace-class, since for a standard basis {ei } of H, T r Cˆµ = i ei · R (Cˆµ ei ) = B µ(dv)kvk2 ≤ 1.) In particular, it is compact, thus admits a basis of orthonormal eigenvectors for H. We write {λi (µ)} for the non-zero eigenvalues. We write Hµ for the eigenspace cooresponding to the non-zero eigenvalues. We have H = Hµ ⊕ Hµ⊥ . It is easily verified that µ(Hµ⊥ ∩ B) = 0. We define dim µ := #{i ∈ N : λi (µ) > 0} = dim Hµ . 32
We claim that the function µ 7→ dim µ is an invariant of the cavity mapping. Standard arguments show that it is measurable with respect to the Borel v-algebra of M(B). It is readily checked that eigenvectors of µ are mapped to eigenvectors of µT −1 for any isometry T of B, therefore dim µ = dim µT −1 . It remains to show that dim µ = dim Φl µ Pl -a.s. We have v ∈ Hµ⊥ ∩ B if and only if Z ˆ v · (Cµ v) = µ(dv) (v, v)2 = 0 . B
By Lemma 4.4, this is true if and only if Z Φl µ(dv) (v, v)2 = v · (CˆΦl µ v) = 0, B
⊥ which in turn is equivalent to v ∈ HΦ ∩ B. The claim follows. lµ The following result on the dimension of stochastically stable ROSt is a generalization of a result for competing particle systems which states that no system with a finite number of particles can be quasi-stationary [25]. It is rather surprising that it holds for stochastic stability under a single cavity field.
Theorem 1.7. Let µ be the sampling measure of a stochastically stable ROSt. Then dim µ = 1 or ∞. In other words, it is supported on a single vector or on an infinite-dimensional subset of B. Proof. By relying on the Choquet decomposition, we can suppose that the law P is in Ext(SS). Since the dimension is an invariant of the mapping, we know by Lemma 4.3 that it is a constant with probability one. Suppose 1 < dim µ < ∞. We first prove that this implies that the sampling measure µ is supported on a sphere P-almost surely, that is, there exists a deterministic r ≥ 0 such that µ{v ∈ B : kvk = r} = 1 for almost all µ. Indeed, define n o rmin(µ) := inf r ≥ 0 : µ{v ∈ B : kvk ≤ r} > 0 . We claim that the function µ 7→ rmin (µ) is an invariant of the cavity mapping; hence it is a constant: rmin := Ermin (µ) = rmin (µ) P-a.s. Indeed it is measurable since we can write o n [ {µ ∈ M(B) : rmin (µ) < s} = µ ∈ M(B) : µ{kvk ≤ s′ } > 0 , s′ ∈Q, 0≤s′ 0 ⇐⇒ Φl µ{v ∈ B : kvk ≤ r} > 0 . This proves that rmin(µ) is constant almost surely. This constant will henceforth be written simply as rmin . For any r > rmin and ǫ > 0, consider the ratio µ{v ∈ B : kvk > r + ǫ} . (4.2) µ{v ∈ B : kvk ≤ r} We claim that the ratio is zero for any r > rmin and ǫ > 0. If so, we are done, since it implies that µ{v ∈ B : kvk > r} = 0 for any r > rmin , hence that µ is supported on the sphere of radius rmin . For any T ∈ N, apply T iterates of the cavity mapping Φl to µ. By stochastic stability, we must have that ratio (4.2) has the same law as √
µ(e
T l(v)− T2 kvk2 ; kvk √ T l(v)− T2 kvk2
> r + ǫ)
; kvk ≤ r)
µ(e
.
Since 1 < dim µ < ∞, l can be seen as a Gaussian vector in a finitedimensional space. We then have the bound −klk ≤ l(v) ≤ klk, since kvk ≤ 1. A straightforward application of that bound and simple estimates shows that the ratio is smaller or equal to √
e2
T klk− T2 (2ǫr+ǫ2 ) µ{v
∈ B : kvk > r + ǫ} , µ{v ∈ B : kvk ≤ r}
and the claim follows for Pl -almost all l by taking the limit T → ∞. It remains to show that if µ has support in a finite-dimensional sphere, then it must be supported on a single vector. The idea is that µ must be supported on the vector where the cavity field is maximal. In a finitedimensional space, v 7→ l(v) is a continuous function for Pl -almost all l. Since the sphere is compact, it achieves its maximum, say ml . Moreover, by independence of the l(v)’s in orthogonal directions, the maximizer, say vl∗ , is unique. Let Cl (ǫ) be a cap neighborhood of the maximizer vl∗ on the sphere such that l(v) > ml − ǫ for v ∈ Cl (ǫ) and l(v) ≤ ml − ǫ for v ∈ Cl (ǫ)c . If the 34
measure µ is not supported on a single vector, there exists an ǫ such that µ(Cl (ǫ)c ) > 0. We then have √ µ e T l(v) ; Cl (ǫ/2) √ µ{Cl (ǫ/2)} √ ≥ e T ǫ/2 . µ{Cl (ǫ)c } µ e T l(v) ; Cl (ǫ)c This shows that, for any ǫ, the limit of the left-hand side as T → ∞ exists and is infinite for P-almost all µ and Pl -almost all l. But by stochastic stability, the ratio on the left-hand side has the same law as for a single application of Φl , i.e., µ el(v) ; Cl (ǫ/2) . µ (el(v) ; Cl (ǫ)c )
This ratio is infinite for all ǫ if and only if (Φl µ){vl∗ } = 1 P × Pl -a.s. We deduce by stochastic stablity that µ must be supported on a single vector. One might expect that other properties of stochastically stable ROSt’s might be obtained in a similar way leading to a better understanding of the hypotheses of the Ultrametricity Conjecture. If stochastic stability is assumed for an infinite number of cavity fields, we can prove more. Theorem 1.8. If a ROSt satisfies stochastic stability for the cavity field lp for an infinite number of p ∈ N, then its sampling measure is supported almost surely on a sphere. Proof. By relying again on the Choquet decomposition, we can suppose that the law P is in the extreme set of stochastic stable ROSt’s for the given infinite collection of p’s. Let rmax be the essential supremum of the norm of the vectors: rmax (µ) = sup{0 ≤ r ≤ 1 : µ{v ∈ B : kvk ≥ r} > 0} . A similar argument as for rmin(µ) in the proof of Theorem 1.7 shows that rmax (µ) is an invariant under any cavity mapping, thus essentially a constant: rmax := Ermax (µ) P-a.s. If rmax = 0, µ must be supported on the zero vector and the conclusion holds for the sphere of radius 0. Suppose now that 35
rmax > 0. Take a sequence of numbers T = T (p) depending on p such that 2p T (p)rmax converges as p → ∞ to 1. (If rmax = 1, this sequence is simply T (p) ≡ 1.) Define for δ < 1 the set Aδ = {v ∈ B : kvk ≤ δ rmax }. We show E[µ(Aδ )] = 0 for any δ < 1. From the stability assumptions, we have that for any p ∈ N √ T (p) 2p µ e T (p) lp (v)− 2 kvk ; Aδ √ . E[µ(Aδ )] = EEl T (p) 2p µ e T (p)lp (v)− 2 kvk
We take the limit p → ∞ on the right-hand side. By definition of T (p) and Aδ , the ratio inside the expectation converges to zero. (Recall that 2p El lp (v)2 = kvk2p ≤ rmax .) The proposition is proved.
We remark that stochastic stability alone is not sufficient to imply that the sampling measure of a ROSt be supported on a sphere. This is demonstrated by the following example. Take p = (pi )i∈N a Poisson-Dirichlet variable with parameter x1 , and p′ = (p′j )j∈N a Poisson-Dirichlet variable with parameter x2 6= x1 . Pick q1 6= q2 strictly positive such that q1 + q2 = 1 and (1 − x1 )q1 = (1 − x2 )q2 .
(4.3)
P It suffices to consider the ROSt with sampling measure µ = i pi δui + P ′ δ where (u ) are orthogonal vectors of norm q and (v ) are orthogonal p i 1 j j j vj vectors with norm q2 also orthogonal to the u’s. A Poisson-Dirichlet variable with parameter x has the property that, for any positive random variable W such that EW x < ∞ and for a iid sequence (Wi ) with the law of W (see e.g. [5]), law (pi Wi ) = (E[W x ]1/x pi ) . Therefore taking Wi = el(ui ) and Wj = el(vj ) , we have for the variables defined above and for the cavity field l: q1 q2 law −(1−x1 )q1 law −(1−x2 )q2 ′ (pi el(ui )− 2 ) = e 2 (pi ) , and (p′j el(vj )− 2 ) = e 2 (pj ) .
The relation (4.3) ensures that the constant is the same in both cases, hence vanishes after normalization of the weights. This shows that the constructed µ is stochastically stable The above setup also provides a counterexample to the Ultrametricity Conjecture 1 if stochastic stability for a single cavity field alone holds. This 36
is demonstrated by the following example coming from the limit of two uncoupled REM’s as studied in [8]. It suffices to consider the ROSt with samP pling measure µ = i,j pi p′j δ √1 (ui +vj ) . It is easily checked that this ROSt is 2 stochastically stable from the invariance property of Poisson-Dirichlet variables and the fact that l( √12 (ui +vj )) = √12 l(ui )+ √12 l(vj ) (the restriction (4.3) is not needed here). On the other hand, the support of its sampling measure is not ultrametric as it can be easily checked from the following triplet of vectors in the support of µ: √12 (u1 + v1 ), √12 (u2 + v2 ) and √12 (u1 + v2 ).
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