arXiv:math-ph/0201012v2 7 Jan 2002
Journ. Statist. Phys. 88 (1997), 269-305
Correlations between Zeros of a Random Polynomial Pavel Bleher1 and Xiaojun Di1 Revised version
Abstract. We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random polynomial.
Key words: Real random polynomials; correlations between zeros; scaling limit; determinants of block matrices.
1
Department of Mathematical Sciences, Indiana University – Purdue University at Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, USA. E-mail:
[email protected],
[email protected]. 1
1. Introduction Let fn (t) be a real random polynomial of degree n, fn (t) = c0 + c1 t + · · · + cn tn ,
(1.1)
where c0 , c1 , . . . , cn are independent real random variables. Distribution of zeros for various classes of random polynomials is studied in the classical papers by Bloch and Polya [BP], Littlewood and Offord [LO], Erd¨os and Offord [EO], Erd¨os and Tur´an [ET], and Kac [K1–K3]. We will assume that the coefficients c0 , c1 , . . . , cn are normally distributed with E c2j = σj2 .
E cj = 0,
(1.2)
In the case when σj2 = 1, fn (t) is the Kac random polynomial. Another interesting case is when n 2 σj = . j As is pointed out by Edelman and Kostlan [EK], “this particular random polynomial is probably the more natural definition of a random polynomial”. We call this polynomial the SO(2) random polynomial because its m-point joint probability distribution of zeros is SO(2)-invariant for all m (see section 5 below). The SO(2) random polynomial can be viewed as the Majorana spin state [Maj] with real random coefficients, and it models a chaotic spin wavefunction in the Majorana representation. See the papers by Leboeuf [Leb1, Leb2], Leboeuf and Shukla [LS], Bogomolny, Bohigas, and Leboeuf [BBL2], and Hannay [Han], where the SU(2) and some other random polynomials are introduced and studied, that represent the Majorana spin states with complex random coefficients. Let {τ1 , . . . , τk } be the set of real zeros of fn (t). Consider the distribution function of the real zeros, Pn (t) = E #{j : τj ≤ t}, where the mathematical expectation is taken with respect to the joint distribution of the coefficients c0 , . . . , cn . Let pn (t) = Pn′ (t) be the density function. By the Kac formula (see, e.g., [K3]), p An (t)Cn (t) − Bn2 (t) . pn (t) = π An (t) 2
(1.3)
where An (t) = Bn (t) = Cn (t) =
n X
j=0 n X
j=1 n X
σj2 t2j , jσj2 t2j−1 =
A′n (t) , 2
j 2 σj2 t2j−2 =
j=1
(1.4)
A′′n (t) A′n (t) + . 4 4t
The derivation of (1.3) by Kac is rather complex. A short proof of (1.3) is given in the paper [EK] by Edelman and Kostlan. See also the papers by Hannay [Han] and Mesincescu, Bessis, Fournier, Mantica, and Aaron [M-A], and section 2 below. The formula (1.3) implies that for the Kac random polynomial, lim pn (t) = p(t) =
n→∞
and
1 , π|1 − t2 |
t 6= ±1,
(1.5)
1/2 1 n(n + 2) pn (±1) = π 12
(see [K3], [BS], and [EK]). The limiting density p(t) is not integrable at ±1, and this means that the zeros are mostly located near ±1. Observe, in addition, that pn (t) is an even function of t, and the distribution pn (t)dt is invariant with respect to the transformation t → 1/t. Kac [K1] proves that the expected number of real zeros has the asymptotics Z ∞ Nn = pn (t) dt = (2/π) log n + O(1). −∞
Kac [K2], Erd¨os and Offord [EO], Stevens [Ste], Ibragimov and Maslova [IM], Logan and Shepp [LS], Edelman and Kostlan [EK], and others extend this asymptotics to various classes of the random coefficients {cj }. Maslova [Mas1] evaluates the variance of the number of real zeros as 2 4 1− ln n(1 + o(1)), n → ∞, Var #{j : fn (τj ) = 0} = π π and she proves the central limit theorem for the number of real zeros (see [Mas2]), for a class of distributions of the random coefficients {cj }. In this paper we are interested in correlations between the zeros τj of the Kac random polynomial. Let us consider first the zeros in the interval (−1, 1). Define straightening of τj as Z t p(u) du. ζj = P (τj ), P (t) = 0
3
In the limit when n → ∞, the straightened zeros ζj are uniformly distributed on the real line, so that lim E # {j : a < ζj ≤ b} = b − a. (1.6) n→∞
From (1.5) we get that P (t) =
Z
hence
t 0
1 + t du 1 = 1 artanh t, = ln 2 π(1 − u ) 2π 1 − t π
1 artanh τj . (1.7) π Let pmn (s1 , . . . , sm ) be the joint probability distribution density of the straightened zeros ζj , ζj =
Pr {∃ ζj1 ∈ [s1 , s1 + ∆s1 ], . . . , ∃ ζjm ∈ [sm , sm + ∆sm ]} . ∆s1 ,...,∆sm →0 |∆s1 . . . ∆sm | (1.8) It coincides with the correlation function E ξn (s1 , s1 + ∆s1 ) . . . ξn (sm , sm + ∆sm ) knm (s1 , . . . , sm ) = lim . (1.9) ∆s1 ,...,∆sm →0 |∆s1 . . . ∆sm | pnm (s1 , . . . , sm ) =
lim
where ξn (a, b) = # {j : a < ζj ≤ b}. We assume in (1.8) and (1.9) that si 6= sj for all i 6= j. Our aim is to find the limit correlation functions km (s1 , . . . , sm ) = lim knm (s1 , . . . , sm ). n→∞
(1.10)
We prove the following results. Theorem 1.1. The limit two-point correlation function k2 (s1 , s2 ) of the straightened zeros ζj = π −1 artanh τj of the Kac random polynomial is equal to k2 (s1 , s2 ) = tanh2 π(s1 − s2 ) +
1 | sinh π(s1 − s2 )| arcsin 2 cosh π(s1 − s2 ) cosh π(s1 − s2 )
(1.11)
Observe that k2 (s1 , s2 ) depends only on s1 − s2 , and it has the following asymptotics: π2 |s1 − s2 | + O(|s1 − s2 |2 ), |s1 − s2 | → 0, 2 16 −4π|s1 −s2 | k2 (s1 , s2 ) = 1 − e + O(e−6π|s1 −s2 | ), |s1 − s2 | → ∞. 3 k2 (s1 , s2 ) =
4
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
s
Fig 1: The two-point correlation function of straightened zeros of the Kac random polynomial.
The graph of k2 (0, s) is given in Fig. 1. Theorem 1.2. The limit m-point correlation function km (s1 , . . . , sm ) of the straightened zeros ζj = π −1 artanh τj is equal to Z ∞ Z ∞ 1 −m (1.12) |y1 . . . ym |e− 2 (Y Γm ,Y ) dy1 . . . dym , ··· km (s1 , . . . , sm ) = 2 −∞
−∞
where Y = (y1 , . . . , ym ) and the matrix Γm is defined as m 1 Γm = cosh π(si − sj ) i,j=1
(1.13)
In particular, km (s1 , . . . , sm ) depends only on the differences of s1 , . . . , sm , hence it is translation invariant. The proof of Theorems 1.1 and 1.2 is given in sections 2, 3 and 4 below. It is based on computation of the determinant of some matrices which consist of 2 × 2 blocks. This computation is of independent interest. The basic example is the matrix ∆11 ∆12 . . . ∆1m ∆22 . . . ∆2m ∆ (1.14) ∆m = 21 ... ... ... ... ∆m1 ∆m2 . . . ∆mm where
1 1 − ti tj ∆ij = tj (1 − ti tj )2
ti (1 − ti tj )2 1 + ti tj , (1 − ti tj )3 5
i, j = 1, . . . , m.
(1.15)
We prove in section 4 that det ∆m = Qm
i=1 (1
8 1≤i 1.
(1.17)
t
In the limit when n → ∞, the straightened zeros ζj are uniformly distributed on the real line. From (1.5) 1 + t 1 = 1 artanh t−1 , P (t) = ln (1.18) 2π 1 − t π so that
ζj = π −1 artanh τj−1 .
(1.19)
out Denote by knm (s1 , . . . , sm ) the correlation function of the straightened zeros ζj with |τj | > 1.
Theorem 1.3. out (s1 , . . . , sm ) = knm (s1 , . . . , sm ). knm
(1.20)
In other words, the correlation functions of the straightened zeros outside of the interval (−1, 1) coincide with those inside of the interval. Finally, let us consider correlation between zeros inside of the interval (−1, 1) and outside of this interval. Let Knm (t1 , . . . , tm ) be the correlation function of the zeros τj (without straightening). Theorem 1.4. Assume that |t1 |, . . . , |tl | < 1 and |tl+1 |, . . . , |tm | > 1. Then the limit lim Knm (t1 , . . . , tm ) = Km (t1 , . . . , tm )
(1.21)
Km (t1 , . . . , tm ) = Kl (t1 , . . . , tl ) Km−l (tl+1 , . . . tm ).
(1.22)
n→∞
exists and
6
This means that the zeros inside and outside of the interval (−1, 1) are asymptotically independent. Observe that Km (t1 , . . . , tm ) km (s1 , . . . , sm ) = , (1.23) p(t1 ) . . . p(tm ) t1 =P −1 (s1 ),...,tm =P −1 (sm ) provided that either all |tj | < 1 or all |tj | > 1 (cf. the formula (2.14) below). Proof of Theorems 1.3 and 1.4 is given in the end of section 4. In sections 5 and 6 we investigate correlation functions of real zeros of the SO(2) random polynomial. 2. General Formulae Let fn (t) =
n X
cj tj ,
(2.1)
j=0
be a polynomial whose coefficients cj are random variables with an absolutely continuous joint distribution. Let ξn (a, b) = #{τk : a < τk ≤ b, fn (τk ) = 0}
(2.2)
be the number of real roots of fn (t) between a and b, and let pn (t) be the density of real zeros tk of fn (t), so that Z b pn (t) dt. (2.3) E ξn (a, b) = a
It is not difficult to show that
pn (t) =
Z
∞
−∞
|y| Dn (0, y; t) dy,
where Dn (x, y; t) is a joint distribution density of fn (t) and fn′ (t), Z bZ d ′ Pr { a < fn (t) ≤ b; c < fn (t) ≤ d } = Dn (x, y; t) dxdy. a
(2.4)
(2.5)
c
Indeed, if fn′ (t) = y then asymptotically as ∆t → 0, the function fn (t) has a zero in the interval [t, t + ∆t] if fn (t) is in the interval [0, −y∆t], and this gives (2.5). Similarly, the m-point correlation function Knm (t1 , . . . , tm ) for pairwise different t1 , . . . , tm is equal to Z ∞ Z ∞ |y1 . . . ym |Dnm (0, y1 , . . . , 0, ym ; t1 , . . . , tm )dy1 . . . dym , ··· Knm (t1 , . . . , tm ) = −∞
−∞
(2.6)
7
where Dnm (x1 , y1 , . . . , xm , ym ; t1 , . . . , tm ) is a joint distribution density of the vector Fn = (fn (t1 ), fn′ (t1 ), . . . , fn (tm ), fn′ (tm )), so that Pr {a1 < fn (t1 ) ≤ b1 ; c1 < fn′ (t1 ) ≤ d1 ; . . . ; am < fn (tm ) ≤ bm ; cm < fn′ (tm ) ≤ dm } Z b1 Z d1 Z bm Z dm = ··· Dnm (x1 , y1 , . . . , xm , ym ; t1 , . . . , tm ) dx1 dy1 . . . dxm dym . a1
c1
am
cm
(2.7) If {cj } are independent random variables with Var cj > 0 then the covariance matrix of the vector Fn is positive, provided that n ≥ 2m − 1 (see Appendix B at the end of the paper). Similar formulae are derived for the correlation functions of complex zeros of random polynomials with complex and real coefficients (see [Han] and [M-A]). Observe that E
m Y
ξn (aj , bj ) =
j=1
Z
b1
a1
···
Z
bm
Knm (t1 , . . . , tm ) dt1 . . . dtm ,
(2.8)
am
provided that (a1 , b1 ), . . . , (am , bm ) are pairwise disjoint, and pn (t) = Kn1 (t),
E (ξn (a, b)) =
Z
b
Kn1 (t)dt.
(2.9)
a
For the general case, when (a1 , b1 ), . . . , (am , bm ) may intersect, we have the following extension of (2.8):
E
m Y
j=1
ξn (aj , bj ) =
X
l Y T
(A1 ,...,Al ) j=1
Z
(ai ,bi )
i∈Aj
dtj Knl (t1 , . . . , tl ),
(2.10)
where the sum is taken over all possible partitions (A1 , . . . , Al ) of {1, . . . , m}, such that Ai ∩ Aj = ∅,
i 6= j,
A1 ∪ · · · ∪ Al = {1, . . . , m}, |Ai | ≥ 1
(2.11)
i = 1, . . . , l.
In particular, when m = 2 we have E [ξn (a1 , b1 )ξn (a2 , b2 )] =
Z 8
b1 a1
Z
b2
a2
Kn2 (t1 , t2 )dt1 dt2
(2.12)
if (a1 , b1 ) ∩ (a2 , b2 ) = ∅, and E
[ξn2 (a, b)]
=
Z
b
pn (t)dt +
a
Z
b a
Z
b
Kn2 (t1 , t2 )dt1 dt2
(2.13)
a
From definition (1.9) of the m-point correlation function, it follows that the m-point correlation function knm (s1 , . . . , sm ) of the straightened zeros ζj = P (τj ) is related to the m-point correlation function Knm (t1 , . . . , tm ) of the zeros τj by the formula
Knm (t1 , . . . , tm ) knm (s1 , . . . , sm ) = p(t1 ) . . . p(tm )
.
(2.14)
t1 =P −1 (s1 ),...,tm =P −1 (sm )
Assume now that the coefficients cj are independent Gaussian variables with zero mean and the variances σj2 , j = 0, . . . , n. Then Dn1 (x, y; t) is a Gaussian distribution density with the covariance matrix E fn2 (t) E fn (t)fn′ (t) An (t) Bn (t) ∆= = , (2.15) E fn (t)fn′ (t) E (fn′ (t))2 Bn (t) Cn (t) where An (t), Bn (t) and Cn (t) are defined in (1.4), and from (2.4) we get the Kac formula (1.3). 3. Two-Point Correlation Function for the Kac Polynomial Let fn (t) = c0 + c1 t + · · · + cn tn be the Kac polynomial, so that ck , k = 0, . . . , n, are real independent Gaussian random variables with E ck 2 = 1.
E ck = 0,
(3.1)
Consider the covariance matrix ∆n of the Gaussian vector (fn (t1 ), fn′ (t1 ), fn (t2 ), fn′ (t2 )). From (3.1) n X 1 − (t1 t2 )n+1 , E fn (t1 )fn (t2 ) = (t1 t2 )k = 1 − t1 t2 k=0 ∂ 1 − (t1 t2 )n+1 ′ (3.2) , E fn (t1 )fn (t2 ) = ∂t1 1 − t1 t2 1 − (t1 t2 )n+1 ∂2 ′ ′ . E fn (t1 )fn (t2 ) = ∂t1 ∂t2 1 − t1 t2 Assume that |t1 |, |t2 | < 1. Then from (3.2) we obtain that lim ∆n = ∆,
n→∞
9
(3.3)
with
∆=
t1 (1−t1 2 )2 1+t1 2 (1−t1 2 )3 1+t1 t2 (1−t1 t2 )3 1+t1 t2 (1−t1 t2 )3
1 1−t1 2 t1 (1−t1 2 )2 1 1−t1 t2 t1 (1−t1 t2 )2
t1 (1−t1 t2 )2 1+t1 t2 (1−t1 t2 )3 t2 (1−t2 2 )2 1+t2 2 (1−t2 2 )3
1 1−t1 t2 t2 (1−t1 t2 )2 1 1−t2 2 t2 (1−t2 2 )2
We prove in the section 4 below that det ∆ =
.
(t1 − t2 )8 . (1 − t1 2 )4 (1 − t2 2 )4 (1 − t1 t2 )8
(3.4)
(3.5)
Let Ω be the two-by-two matrix obtained by removing the first and the third rows and columns from ∆−1 . Then A B Ω= (3.6) B C where
A = (1 − t1 t2 )4 (1 − t1 2 )3 /(t1 − t2 )4
B = (1 − t1 t2 )3 (1 − t1 2 )2 (1 − t2 2 )2 /(t1 − t2 )4 4
2 3
C = (1 − t1 t2 ) (1 − t2 ) /(t1 − t2 )
(3.7)
4
By (2.6), the correlation function K2 (t1 , t2 ) is equal to 1 √ K2 (t1 , t2 ) = 2 4π det ∆
Z
∞ −∞
Z
∞
−∞
1
|y1 y2 |e− 2 (Y Ω,Y ) dy1 dy2
(3.8)
where Y = (y1 , y2 ). Since Z
∞
−∞
Z
∞
1
2
2
|y1 y2 |e− 2 (Ay1 +2By1 y2 +Cy2 ) dy1 dy2 −∞ δ 4 arcsin δ , 1+ √ = AC(1 − δ 2 ) 1 − δ2
B δ=√ AC
(3.9)
(see Appendix A), we obtain that K2 (t1 , t2 ) =
(t1 − t2 )2 π 2 (1 − t1 t2 )2 (1 − t1 2 )(1 − t2 2 )
|t1 − t2 | p arcsin + π 2 (1 − t1 t2 )2 (1 − t1 2 )(1 − t2 2 )
p (1 − t1 2 )(1 − t2 2 ) 1 − t1 t2
(3.10)
Consider the correlation function k2 (s1 , s2 ) of the straightened zeroes ζj = π −1 artanh τj . By (2.14), K2 (t1 , t2 ) k2 (s1 , s2 ) = , t1 = tanh(πs1 ), t2 = tanh(πs2 ). (3.12) p(t1 )p(t2 ) 10
Since p(t) =
1 , π(1 − t2 )
(see (1.5)), we obtain that p p |t1 − t2 | (1 − t1 2 )(1 − t2 2 ) (1 − t1 2 )(1 − t2 2 ) (t1 − t2 )2 + arcsin k2 (s1 , s2 ) = (1 − t1 t2 )2 (1 − t1 t2 )2 1 − t1 t2 | sinh π(s1 − s2 )| 1 = tanh2 π(s1 − s2 ) + arcsin 2 cosh π(s1 − s2 ) cosh π(s1 − s2 ) Theorem 1.1 is proved. 4. Higher order correlation functions for the Kac polynomial Let fn (t) be the Kac polynomial, and let t1 , t2 , . . . , tm be m ≥ 3 distinct points in the (n) interval (−1, 1). Denote by ∆m the covariance matrix of the Gaussian vector (fn (t1 ), fn′ (t1 ), . . . , fn (tm ), fn′ (tm )), (n)
and by ∆m the limit of ∆m as n → ∞, ∆m = lim ∆(n) m
(4.1)
n→∞
Then ∆(n) m
where (n) ∆ij
=
(n)
∆11 ∆(n) 21 = ... (n) ∆m1
(n)
∆12 (n) ∆22 ... (n) ∆m2
... ... ... ...
(n) ∆1m (n) ∆2m ... (n) ∆mm ′
E fn (ti )fn (tj ) E fn (ti )fn (tj ) ′ E fn (ti )fn (tj ) E fn′ (ti )fn′ (tj )
(4.2)
(4.3)
and by (3.2), ∆m where
∆11 ∆ = 21 ... ∆m1
∆ij =
∆12 ∆22 ... ∆m2
1 1−ti tj tj (1−ti tj )2
[cf. (3.4)]. 11
... ... ... ...
∆1m ∆2m ... ∆mm
ti (1−ti tj )2 1+ti tj (1−ti tj )3
!
(4.4)
(4.5)
If Ωm denotes the m × m matrix obtained by removing all the odd number rows and columns from ∆−1 m , then by (2.6), the correlation function Km (t1 , . . . , tm ) is equal to Z ∞ Z ∞ 1 1 √ |y1 . . . ym |e− 2 (Y Ωm ,Y ) dy1 . . . dym (4.6) ··· Km (t1 , . . . , tm ) = m (2π) det ∆m −∞ −∞ where Y = (y1 , . . . , ym ). We have the following extension of the formula (3.6). Proposition 4.1 det ∆m = Qm
i=1 (1
8 1≤i
