Markov property of determinantal processes with extended sine, Airy

arXiv:1106.4360v1 [math.PR] 22 Jun 2011

Markov property of determinantal processes with extended sine, Airy, and Bessel kernels Makoto Katori ∗and Hideki Tanemura

†

22 June 2011

Abstract When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process P are determinantal diffusion processes for any deterministic initial configuration ξ = j∈Λ δxj , in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function Φ having zeros in supp ξ. Using such entire functions Φ, we define new topologies called the Φ-moderate topologies. Then we construct three infinite-dimensional determinantal processes, as the limits of sequences of determinantal diffusion processes with finite numbers of particles in the sense of finite dimensional distributions in the Φ-moderate topologies, so that the probability distributions are continuous with respect to initial configurations ξ with ξ(R) = ∞. We show that our three infinite particle systems are versions of the determinantal processes with the extended sine, Bessel, and Airy kernels, respectively, which are reversible with respect to the determinantal point processes obtained in the bulk scaling limit and the soft-edge scaling limit of the eigenvalue distributions of the Gaussian unitary ensemble, and the hard-edge scaling limit of that of the chiral Gaussian unitary ensemble studied in the random matrix theory. Then Markovianity is proved for the three infinitedimensional determinantal processes. Keywords Determinantal processes · Correlation kernels · Random matrix theory · Infinite particle systems · Markov property · Entire function and topology Mathematics Subject Classification (2010) 15B52, 30C15, 47D07, 60G55, 82C22

∗

Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail: [email protected] † Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan; e-mail: [email protected]

1

1

Introduction

Let M be the space of nonnegative integer-valued Radon measures on R, which is a Polish space with the vague topology: let C0 (R) be the set of all continuous real-valued functions R with compact supports, and we say ξn , n ∈ N converges to ξ vaguely, if limn→∞ R ϕ(x)ξn (dx) = R ϕ(x)ξ(dx) for any ϕ ∈ C0 (R). Each element ξ of M can be represented as ξ(·) = R P j∈Λ δxj (·) with an index set Λ and a sequence of points x = (xj )j∈Λ in R satisfying ξ(K) = ♯{xj : xj ∈ K} < ∞ for any compact subset K ⊂ R. We call an element ξ of M an unlabeled configuration, and a sequence of points x a labeled configuration. A probability measure on a configuration space is called a determinantal point process or Fermion point process, if its correlation functions are generally represented by determinants [32, 33]. In the present paper we say that an M-valued process Ξ(t) is determinantal, if the multitime correlation functions for any chosen series of times are represented by determinants. In other words, a determinantal process is an M-valued process such that, for any integer M ∈ N = {1, 2, . . . }, f = (f1 , f2 , . . . , fM ) ∈ C0 (R)M , a sequence of times t = (t1 , t2 , . . . , tM ) 1, 1 ≤ m ≤ M, the moment with 0 < t1 < · · · < tM < ∞, if we set χtm (x) = efm (x) h −n oi PM R t generating function of multitime distribution, Ψ [f ] ≡ E exp f (x)Ξ(t , dx) , m m=1 R m is given by a Fredholm determinant h i = Det δ δ(x − y) + K(s, x; t, y)χ (y) , (1.1) Ψt [f ] = Det st t 2 (s,t)∈{t1 ,t2 ,...,tM } , (x,y)∈R2

with a locally integrable function K called a correlation kernel [14, 16]. Finite- and infinitedimensional determinantal processes were introduced as multi-matrix models [7, 22], tiling models [11], and surface growth models [30], and they have been extensively studied. In the present paper we study the infinite-dimensional determinantal processes describing onedimensional infinite particle systems with long-range repulsive interactions. They are obtained by taking appropriate N → ∞ limits of the N-particle systems of noncolliding diffusion processes, which dynamically simulate the eigenvalue statistics of the Gaussian random matrix ensembles studied in the random matrix theory [21, 8]. The purpose of the present paper is to prove Markov property of the three infinite-dimensional determinantal processes with the correlation kernels called the extended sine, Airy, and Bessel kernels, which are reversible with respect to the probability measures obtained in the bulk scaling limit and the soft-edge scaling limit of the eigenvalue distribution in the Gaussian unitary ensemble (GUE), and in the hard-edge scaling limit of that in the chiral Gaussian unitary ensemble (chGUE), respectively. See, for instance, [19] and the references therein. Dyson [6] introduced a stochastic model of particles in R with a log-potential, which obeys the stochastic differential equations (SDEs): dXj (t) = dBj (t) +

X

dt , Xj (t) − Xk (t) 1≤k≤N,k6=j

1 ≤ j ≤ N,

t ∈ [0, ∞),

(1.2)

where Bj (t)’s are independent one-dimensional standard Brownian motions. This is a special case with the parameter β = 2 of Dyson’s Brownian motion models [6, 21] but we 2

will call it simply the Dyson model in this paper. It is equivalent to a system of onedimensional Brownian motions conditioned never to collide with each other [19]. For the solution Xj (t), j = 1, 2, . . . , N of (1.2) with initial values Xj (0) = xj , j = 1, 2, . . . , N, we PN P N put Ξ(t, ·) = N j=1 δxj (·). The (unlabeled) Dyson model starting j=1 δXj (t) (·) and ξ (·) = N from ξ N is denoted by ({Ξ(t)}t∈[0,∞) , Pξ ). In the previous paper [16] we showed that for any N fixed unlabeled configuration ξ N , the process ({Ξ(t)}t∈[0,∞) , Pξ ) is determinantal with the N correlation kernel Kξ specified by an entire function Π0 , which is explicitly given by (2.1) below in the present paper. Then for configurations ξ ∈ M with ξ(R) = ∞, a sufficient condition was given so that the sequence of the processes ({Ξ(t)}t∈[0,∞), Pξ∩[−L,L] ) converges to an M-valued determinantal process as L → ∞ in the sense of finite dimensional distributions in the vague topology, where ξ ∩ [−L, L] denotes the restricted measure of ξ on an interval [−L, L]. Note that ξ ∈ M implies ξ ∩ [−L, L](R) < ∞ for 0 < L < ∞. The limit process is the Dyson model with an infinite number of particles and denoted by ({Ξ(t)}t∈[0,∞), Pξ ). Recently, in [18] the tightness of the sequence of processes was proved. The class of configurations satisfying our condition, denoted by Y, is large enough to carry the Poisson point processes, Gibbs states with regular conditions, as well as the determinantal (Fermion) point process µsin with the sine kernel: Z sin{π(y − x)} 1 dk eik(y−x) = , x, y ∈ R, (1.3) Ksin(x, y) ≡ 2π |k|≤π π(y − x)

√ where i = −1. From the uniqueness of solutions of (1.2), the Dyson model with a finite number of particles is a diffusion process (i.e. it is a strong Markov process having a continuous path almost surely). Although the process ({Ξ(t)}t∈[0,∞) , Pξ ) is given as the limit of a sequence of diffusion processes ({Ξ(t)}t∈[0,∞) , Pξ∩[−L,L]), L ∈ N, the Markov property could be lost in it. In the Dyson model, because of the long-range interaction, if the number of particles is infinite, the probability distribution Pξ is not vaguely continuous with respect to the initial configuration ξ. Therefore the Markovianity of the process is not readily concluded in the infinite-particle limit of a sequence of Markov processes R in the vague topology. For a probability measure µ on M we put Pµ (·) = M µ(dξ)Pξ (·). Suppose that ξ(R) = ∞, µ-almost surely. For proving Markov property of the process ({Ξ(t)}t∈[0,∞) , Pµ ), it is sufficient b of M with a topology T , which is stronger than the vague topology, and to find a subset Y a sequence of probability measures {µN }N ∈N such that 1. Pξ (·) is continuous with respect to ξ in the topology T , b = 1, for any t ∈ [0, ∞), 2. Pµ (Ξ(t) ∈ Y)

b : η(R) = N}) = 1, N ∈ N, 3. µN ({η ∈ Y

4. ({Ξ(t)}t∈[0,∞) , PµN ) converges ({Ξ(t)}t∈[0,∞) , Pµ ) as N → ∞ in the sense of finite dimensional distributions in the topology T . 3

In this paper first we will show that these conditions are satisfied in the case that µ = µsin , b with the topology T called the Φ0 -moderate topology defined by (2.2) given if we use Y as Y below. We also show that the process ({Ξ(t)}t∈[0,∞) , Pµsin ) is a version of the determinantal process ({Ξ(t)}t∈[0,∞) , Psin ) associated with the extended sine kernel Ksin with density 1: Z 1 2 Ksin (s, x; t, y) ≡ dk ek (t−s)/2+ik(y−x) − 1(s > t)p(s − t, x|y) 2π |k|≤π  Z 1  2 2   du eπ u (t−s)/2 cos{πu(y − x)} if t > s   0 = (1.4) Ksin y) if t = s Z (x,  ∞  2 2   du eπ u (t−s)/2 cos{πu(y − x)} if t < s.  − 1

(Two processes having the same state space are said to be equivalent if they have the same finite-dimensional distributions, and we also say that each one is a version of other or they are versions of the same process. See Section 1.1 of [31].) Hence, we conclude that the determinantal process with the extended sine kernel is a Markov process which is reversible with respect to the determinantal point process µsin (Theorem 2.6). The above strategy will be also used to show Markovianity of the infinite-dimensional determinantal process ({Ξ(t)}t∈[0,∞) , PJν ) associated with the extended Bessel kernel KJν :  Z 1 √ √    if s < t du e−2u(s−t) Jν (2 ux)Jν (2 uy)    0     √ √ √ √ √ √  Jν (2 x) yJν′ (2 y) − xJν′ (2 x)Jν (2 y) KJν (s, x; t, y) = (1.5) if t = s  x − y      Z ∞   √ √   du e−2u(s−t) Jν (2 ux)Jν (2 uy) if s > t,  − 1

x, y ∈ R+ ≡ {x ∈ R : x ≥ 0}, where Jν (·) is the Bessel function with index ν > −1 defined by ∞  z 2n+ν X (−1)n Jν (z) = , z∈C Γ(n + 1)Γ(n + 1 + ν) 2 n=0 R∞ with the gamma function Γ(x) = 0 e−u ux−1du (Theorem 2.7). This process is reversible with respect to the determinantal point process µJν with the Bessel kernel  √ √ √ √ √ √  Jν (2 x) yJν′ (2 y) − xJν′ (2 x)Jν (2 y) if x 6= y, KJν (x, y) = (1.6) x − y √ √ √  2 (Jν (2 x) − Jν+1 (2 x)Jν−1 (2 x) if x = y, with Jν′ (x) = dJν (x)/dx.

4

Finally we will study the infinite-dimensional determinantal process ({Ξ(t)}t∈[0,∞) , PAi ) associated with the extended Airy kernel KAi :  Z ∞   du e−u(t−s)/2 Ai(u + x)Ai(u + y) if t ≥ s  Z0 0 KAi (s, x; t, y) ≡ (1.7)  −u(t−s)/2  du e Ai(u + x)Ai(u + y) if t < s,  − −∞

x, y ∈ R, where Ai(·) is the Airy function defined by Z 1 3 Ai(z) = dk ei(zk+k /3) , 2π R

z ∈ C,

which is reversible with respect to the determinantal point process µAi with the Airy kernel  ′ ′  Ai(x)Ai (y) − Ai (x)Ai(y) if x 6= y (1.8) KAi (x, y) = x−y  ′ 2 2 (Ai (x)) − x(Ai(x)) if x = y,

where Ai′ (x) = dAi(x)/dx. In this process the density ρAi (x) of particles is not bounded √ and shows the same asymptotic behavior with the function ρb(x) ≡ ( −x/π)1(x < 0) as x → −∞, where 1(ω) is the indicator function of a condition ω; 1(ω) = 1 if ω is satisfied, and 1(ω) = 0 otherwise (see (4.12) in Lemma 4.3). Therefore the repulsive force from infinite number of particles in the negative region x < 0 causes a positive drift with infinite strength. To compensate it in order to obtain a stationary system, another negative drift with infinite strength should be included in the process. For this reason we have to modify our strategy and we introduce not only a sequence of probability measures but also a sequence of N approximate processes with finite numbers of particles ({ΞρbN (t)}t∈[0,∞) , Pξ ), N ∈ N, having negative drifts such that their strength diverges as N goes to infinity. To determine the Ndependence of negative drifts , we use the estimate (4.13) in Lemma 4.3. Then we will show that the limiting process is a version of ({Ξ(t)}t∈[0,∞) , PAi) and it is Markovian (Theorem 2.8). Remark 1. Other processes having infinite-dimensional determinantal point processes as their stationary measures have been constructed and Markovianity of systems was proved in Borodin and Olshanski [3, 4, 5], Olshanski [24], and Borodin and Gorin [2]. They determined the state spaces and transition functions associated with Feller semi-groups and concluded that these infinite-dimensional determinantal processes are strong Feller processes. In the present paper, we first prove Markovianity of the determinantal processes with the extended sine kernel Ksin, the extended Airy kernel KAi , and the extended Bessel kernel KJν , which have been well-studied in the random matrix theory and its related fields. To these processes, the argument by [3, 4, 5, 24, 2] can not be applied, and the strong Feller property has not yet proved. (See also Remark 2 given at the end of Section 2.) The paper is organized as follows. In Section 2 preliminaries and main results are given. In Section 3 the basic properties of the correlation functions are summarized. Section 4 is devoted to proofs of results by using Lemma 4.3. In Section 5 Lemma 4.3 is proved by using the estimate of Hermite polynomials given by Plancherel and Rotach [29]. 5

2

Preliminaries and Main Results

2.1

Non-equilibrium dynamics

For ξ N ∈ M with ξ N (R) = N ∈ N and p ∈ N0 ≡ N ∪ {0} we consider the product  w ξ({x}) Y N Πp (ξ , w) = G ,p , w ∈ C, x N x∈supp ξ

where G(u, p) =

 1 − u,   

if p = 0

  p 2 u u   , if p ∈ N. +···+  (1 − u) exp u + 2 p

The functions G(u, p) are called the Weierstrass primary factors [20]. Then we set ξ({x})  Y w−z N N c ,p , w, z ∈ C, Φp (ξ , z, w) ≡ Πp (τ−z ξ ∩ {0} , w − z) = G x−z N c x∈supp ξ ∩{z}

P

P where τz ξ(·) ≡ j∈Λ δxj +z (·) for z ∈ C and ξ(·) = j∈Λ δxj (·). N It was proved in Proposition 2.1 of [16] that the Dyson model ({Ξ(t)}t∈[0,∞) , Pξ ), starting N from any fixed configuration ξ N ∈ M is determinantal with the correlation kernel Kξ given by Z I Π0 (τ−z ξ N , iu − z) 1 ξN du dz psin (s, x|z) psin (−t, iu|y) K (s, x; t, y) = 2πi R iu − z Ciu (ξ N ) −1(s > t)psin (s − t, x|y), (2.1) where Cw (ξ N ) denotes a closed contour on the complex plane C encircling the points in supp ξ N on the real line R once in the positive direction but not the point w, and psin (t, x|y) is the generalized heat kernel: n (x − y)2 o 1 psin (t, x|y) = p exp − 1(t 6= 0) + δ(y − x)1(t = 0), t ∈ R, x, y ∈ C. 2t 2π|t|

We put M0 = {ξ ∈ M : ξ(x) ≤ 1 for any x ∈ R}. Any element ξ ∈ M0 has no multiple points and can be identified with its support which is a countable subset of R. In case ξ N ∈ M0 , (2.1) is rewritten as Z Z ξN N ′ K (s, x; t, y) = ξ (dx ) du psin(s, x|x′ )Φ0 (ξ N , x′ , iu)psin(−t, iu|y) R

R

−1(s > t)psin (s − t, x|y).

For ξ ∈ M, p ∈ N0 , w, z ∈ C we define Φp (ξ, z, w) = lim Φp (ξ ∩ [−L, L], z, w) L→∞

6

if the limit finitely exists. For L > 0, α > 0 and ξ ∈ M we put Z ξ(dx) M(ξ, L) = , and M(ξ) = lim M(ξ, L), L→∞ x [−L,L]\{0} if the limit finitely exists, and for α > 0 we put Mα (ξ) =

Z

{0}c

1 ξ(dx) |x|α

1/α

.

It is readily to see that Φp (ξ, z, w) finitely exists and is not identically 0 if Mp+1 (ξ) < ∞, and that |Φ0 (ξ, z, w)| < ∞ if |M(ξ)| < ∞ and M2 (ξ) < ∞. P For κ > 0, we put g κ (x) = sgn(x)|x|κ , x ∈ R, and η κ (·) = ℓ∈Z δgκ (ℓ) (·). For κ ∈ (1/2, 1) and m ∈ N we denote by Yκ,m the set of configurations ξ satisfying the following conditions (C.I) and (C.II): (C.I)

|M(ξ)| < ∞,

 m(ξ, κ) ≡ max ξ [g (k), g (k + 1)] ≤ m.

(C.II)

κ

κ

k∈Z

And we put

Y=

[

[

Yκ,m.

κ∈(1/2,1) m∈N

Noting that the set {ξ ∈ M : m(ξ, κ) ≤ m} is relatively compact with the vague topology for each κ ∈ (1/2, 1) and m ∈ N, we see that Y is locally compact. We introduce the following topology on Y. Definition 2.1 Suppose that ξ, ξn ∈ Y, n ∈ N. We say that ξn converges Φ0 -moderately to ξ, if lim Φ0 (ξn , i, ·) = Φ0 (ξ, i, ·) uniformly on any compact set of C. (2.2) n→∞

It is easy to see that (2.2) is satisfied, if ξn converges to ξ vaguely and the following two conditions hold: (2.3) lim sup M(ξn ) − M(ξn , L) = 0, L→∞ n>0 (2.4) lim sup M2 (ξn ∩ [−L, L]c ) = 0. L→∞ n>0 Note that for any a ∈ R and z ∈ C lim Φ0 (ξn , a, z) = Φ0 (ξ, a, z), if ξn converges Φ0 n→∞

moderately to ξ and a ∈ / supp ξ. We denote the space of M-valued continuous functions defined on [0, ∞) by C([0, ∞) → M). We have obtained the following results. (See Theorem 2.4 of [16] and Theorem 1.4 of [18].)

7

Proposition 2.2 (i) If ξ ∈ Y, the process ({Ξ(t)}t∈[0,∞) , Pξ∩[−L,L]) converges to the determinantal process with a correlation kernel Kξ as L → ∞, weakly on the path space C([0, ∞) → M). In particular, when ξ ∈ Y0 ≡ Y ∩ M0 , Kξ is given by Z Z ξ ′ K (s, x; t, y) = ξ(dx ) du psin(s, x|x′ )Φ0 (ξ, x′ , iu)psin(−t, iu|y) R

R

−1(s > t)psin (s − t, x|y).

(ii) Suppose that ξ, ξn ∈ Yκm , n ∈ N, for some κ ∈ (1/2, 1) and m ∈ N. If ξn converges Φ0 -moderately to ξ, then the process ({Ξ(t)}t∈[0,∞) , Pξn ) converges to the process ({Ξ(t)}t∈[0,∞) , Pξ ) as n → ∞, weakly on the path space C([0, ∞) → M). Let M+ = {ξ∩R+ : ξ ∈ M}. We consider a one-parameter family of M+ -valued processes with a parameter ν > −1, X (2.5) Ξ(ν) (t, ·) = δX (ν) (t) (·), t ∈ [0, ∞), j

j∈Λ

(ν)

where Xj (t)’s satisfy the SDEs, q X (ν) (ν) (ν) dXj (t) = 2 Xj (t)dBj (t) + 2(ν + 1)dt + 4Xj (t) j ∈ Λ,

t ∈ [0, ∞)

1 (ν)

(ν)

k:k6=j Xj (t) − Xk (t)

dt, (2.6)

and if −1 < ν < 0 with a reflection wall at the origin. For a given configuration ξ N ∈ M+ of finite particles, ξ N (R+ ) = N ∈ N, the process starting from ξ N is denoted by N ({Ξ(ν) (t)}t∈[0,∞) , Pξ ) and called the noncolliding squared Bessel processes with index ν. In N Theorem 2.1 of [17] it was proved that ({Ξ(ν) (t)}t∈[0,∞) , Pξ ) is determinantal with the correlation kernel Z 0 I 1 Π0 (τ−z ξ N , u − z) (ν) ξN Kν (s, x; t, y) = du dz p(ν) (s, x|z) p (−t, u|y) 2πi −∞ u−z Cu (ξ N ) −1(s > t)p(ν) (s − t, x|y),

(2.7)

where for t ∈ R and x, y ∈ C  √   xy 1  y ν/2 x+y (ν) p (t, y|x) = Iν 1(t 6= 0, x 6= 0) exp − 2|t| x 2t |t|  y yν + 1(t 6= 0, x = 0) + δ(y − x)1(t = 0). (2.8) exp − (2|t|)ν+1 Γ(ν + 1) 2t

Note that for t ≥ 0 and x, y ∈ R+ , p(ν) (t, y|x) is the transition density function of 2(ν + 1)dimensional squared Bessel process. In case ξ N ∈ M+ ∩ M0 , (2.7) is equal to Z ∞ Z 0 ξN N ′ Kν (s, x; t, y) = ξ (dx ) du p(ν) (s, x|x′ )Φ0 (ξ N , x′ , u)p(ν) (−t, u|y) 0

−∞

(ν)

−1(s > t)p (s − t, x|y) 8

+ For κ ∈ (1, 2) and m ∈ N, let Y+ κ,m be the set of configurations ξ of M satisfying (C.II), and put [ [ Y+ Y+ = κ,m . κ∈(1,2) m∈N

Note that Y+ is locally compact in the vague topology. The following proposition is a slight modification of the results stated in Section 2 of [17], which can be proved by the same procedure given there. Proposition 2.3 (i) If ξ ∈ Y+ , the process ({Ξ(ν) (t)}t∈[0,∞) , Pξ∩[−L,L]) converges to the determinantal process with a correlation kernel Kξν as L → ∞ in the sense of finite dimensional + ξ distributions. In particular, when ξ ∈ Y+ 0 ≡ Y ∩ M0 , Kν is given by Z ∞ Z 0 ξ ′ Kν (s, x; t, y) = ξ(dx ) du p(ν) (s, x|x′ )Φ0 (ξN , x′ , u)p(ν) (−t, u|y) 0

−∞ (ν)

−1(s > t)p (s − t, x|y)

(ii) Suppose that ξ, ξn ∈ Y+ κ,m , n ∈ N, for some κ ∈ (1, 2) and m ∈ N. If ξn converges Φ0 -moderately to ξ, then the process ({Ξ(ν) (t)}t∈[0,∞) , Pξn ) converges to the process ({Ξ(ν) (t)}t∈[0,∞) , Pξ ) as n → ∞ in the sense of finite dimensional distributions. Let RρbN , N ∈ N be a sequence of nonnegative functions on R such that ρbN (x) = 0 for √ x ≥ 0, R dx ρbN (x) = N and ρbN (x) ր ρb(x) = ( −x/π)1(x < 0), N → ∞. For ξ N ∈ M with ξ N (R) = N we put Z ρbN (x)dx − ξ N (dx) N MρbN (ξ ) = . x {0}c

and define

"

#

ΦρbN (ξ N , w) ≡ exp wMρbN (ξ N ) Π1 (ξ N ∩ {0}c , w), ΦρbN (ξ N , z, w) ≡ ΦρbN (τ−z ξ N , w − z),

z ∈ C,

w, z ∈ C.

For ξ ∈ M with ξ(R) = ∞ we put MA (ξ) = lim MρbN (ξ N ), if ξ N converges to ξ, N → ∞, N →∞

with the vague topology and

lim sup

L→∞ N ∈N

Z

|x|>L

ρbN (x)dx − ξ N (dx) = 0, x

is satisfied. Then MA (ξ) is also represented as Z MA (ξ) = lim L→∞

0 0 and L1 > 0 such that Z L Z 0 ε ξ([0, L]) − ≤ C1 L , ξ([−L, 0)) − ≤ C1 Lε , L ≥ L1 . ρ(x)dx (3.2) ρ(x)dx 0

−L

(i) If

lim

Z

ρ(x)dx x

finitely exits,

(3.3)

then ξ satisfies (C.I). (ii) If a nonnegative function ρ on R satisfies Z ρ(x)dx − ρ(x)dx −δ ≤ C2 L , x |x|≥L

(3.4)

L→∞

1≤|x|≤L

for some δ > 0, then ξ satisfies Z ρ(x)dx − ξ(dx) ≤ 3C1 Lε−1 + C2 L−δ . 1−ε x |x|≥L

(3.5)

Proof. By using the integration by parts formula with (3.2), we see that for L2 > L ≥ L1 Z L2 Z L2 Z L2 C1 ξ(dx) ρ(x)dx ε−1 ≤ 2C1 L + C1 xε−2 dx ≤ − (3Lε−1 − L2 ε−1 ). x x 1−ε L

L

L

Similarly, we have

Then for L ≥ L1

Z

−L

−L2

ρ(x)dx − x

Z

Z

−L

−L2

ξ(dx) − x |x|≥L

Then (i) and (ii) are derived easily.

C1 ξ(dx) ≤ (3Lε−1 − L2 ε−1 ). x 1−ε

Z

3C1 ε−1 ρ(x)dx ≤ L . x 1−ε |x|≥L 15

Lemma 3.3 Let µ be a probability measure on M with correlation functions ρm , m ∈ N. Suppose that there exist m′ ∈ N and p < m′ − 1 such that Z Z L m′ (3.6) µ(dξ) ξ([0, L)) − ρ(x)dx = O(Lp ), L → ∞, M 0 Z Z 0 m′ (3.7) µ(dξ) ξ([−L, 0)) − ρ(x)dx = O(Lp ), L → ∞. −L

M

(i) In case (3.3) holds and

XZ k∈Z

[g κ (k),g κ (k+1)]m

ρm (xm )dxm < ∞

(3.8)

is satisfied for some m ∈ N and κ ∈ (1/2, 1), then µ(Y) = 1. (ii) In case µ(M+ ) = 1 and (3.8) is satisfied for some m ∈ N and κ ∈ (1, 2), then µ(Y+ ) = 1. (iii) In case (3.4) holds with ρ = ρb and (3.8) satisfied for some m ∈ N and κ ∈ (1/2, 2/3), then µ(YA ) = 1. Proof. Take ε ∈ ((p + 1)/m′ , 1). Using Chebyshev’s inequality with (3.6) and (3.7), we can find a positive constant C such that   Z L ′ ε µ |ξ((0, L]) − ρ(x)dx| ≥ CL ≤ CLp−m ε , 0

and

 Z µ |ξ([−L, 0)) −

0

−L

′

ε

ρ(x)dx| ≥ CL

Since p − m ε < −1, we have

Z X  µ |ξ((0, L]) − L∈N

′

≤ CLp−m ε

L ε

0

Z X  µ |ξ([−L, 0)) − L∈N



ρ(x)dx|)| ≥ CL



0 ε

−L

ρ(x)dx| ≥ CL



< ∞, < ∞,

and so the condition (3.2) is satisfied µ-a.s. ξ by the Borel-Cantelli lemma. Since we have assumed the condition (3.3) in (i), we can apply Lemma 3.2 and conclude that (C.I) holds for µ-a.s. ξ. Similarly, since the condition (3.4) with ρ = ρb is provided in (iii), we conclude that (C.I-A) holds for µ-a.s. ξ. The condition (3.8) implies  X  µ ξ(g κ (k), g κ (k + 1)) > m < ∞. k∈Z

Then, by the Borel-Cantelli lemma again, (C.II) is derived. Therefore, we obtain the lemma.

16

3.2

Equilibrium dynamics

The Dyson model starting from N points all at the origin is determinantal with the correlation kernel      N −1  k/2  1 X t x y   ϕk √ ϕk √ if s ≤ t   √2s s 2s 2t k=0 (3.9) KN (s, x; t, y) =     ∞  k/2 X  y x −1 t   ϕk √ if s > t, ϕk √   √2s s 2s 2t k=N

where hk =

√

π2k k! and

1 2 ϕk (x) = √ e−x /2 Hk (x), hk

(3.10)

with the Hermite polynomials Hk , k ∈ N0 (see Eynard and Mehta [7]). The distribution of the process at time t > 0 is identical with µGUE N,t , the distribution of eigenvalues in the GUE with variance t. The extended sine kernel Ksin with density 1 is derived as the bulk scaling limit [22]:   2N 2N + s, x; 2 + t, y → Ksin(s, x; t, y), N → ∞, (3.11) KN π2 π

which implies

({Ξ(t)}t∈[0,∞) , PµGUE

f.d.

N,2N/π 2

) −→ ({Ξ(t)}t∈[0,∞) , Psin),

N → ∞,

in the vague topology,

(3.12)

f.d.

where −→ means the convergence in the sense of finite dimensional distributions. The noncolliding squared Bessel process starting from N points all at the origin is determinantal with the correlation kernel  N −1  k x y  1 X t  ν  ϕνk , if s ≤ t, ϕ  k  2s s 2s 2t (ν) k=0 KN (s, x; t, y) = ∞  k x y X  1 t  ν  ϕνk , if s > t, ϕk   − 2s s 2s 2t k=N

p where ϕνk (x) = Γ(k + 1)/Γ(ν + k + 1)xν/2 Lνk (x)e−x/2 with the Laguerre polynomials Lνk (x), k ∈ (ν) N0 with parameter ν > −1 [13]. Let µN,t be the distribution of the process at time t > 0. (ν) In the case that ν ∈ N0 , the probability measure µN,t is identical with the distribution of eigenvalues in the chGUE with variance t. The extended Bessel kernel KJν is derived as the hard edge scaling limit [9, 36]: (ν)

KN (N + s, x; N + t, y) → KJν (s, x; t, y). N → ∞, 17

(3.13)

which implies f.d.

({Ξ(ν) (t)}t∈[0,∞) , Pµ(ν) ) −→ ({Ξ(ν) (t)}t∈[0,∞) , P(ν) ), N,N

N → ∞,

in the vague topology.

(3.14)

The extended Airy kernel is derived as the soft-edge scaling limit [30, 12, 23]:   s2 t2 KN N 1/3 + s, 2N 2/3 + N 1/3 s − + x; N 1/3 + t, 2N 2/3 + N 1/3 t − + y 4 4 → KAi (s, x; t, y), N → ∞. (3.15) √ We introduce the nonnegative function fsc (x) = (2/π) 1 − x2 1(|x| ≤ 1), and put r   √ x 1 √ N N √ fsc ρsc (t, x) = 4tN − x2 1(|x| ≤ 2 tN ), = 4t 2πt 2 Nt associated with Wigner’s semicircle law of eigenvalue distribution in the GUE [21]. Then we put x  N 1/3  N 1/3 2/3 f 1 + 1(x < 0) ρbN (x) = ρ (N , 2N + x) = sc sc sc 2 2N 2/3 r  x  1 −x 1 + = 1(x < 0). π 4N 2/3

Here we note that ρbN b(x), N → ∞ and sc (x) ր ρ Z Z N dx ρbsc (x) = N, and

0

dx

−4N 2/3

R

Then (3.15) implies

f.d.

({ΞρbNsc (t)}t∈[0,∞) , Pτ

µGUE −2N 2/3 N,N 1/3

4 4.1

ρbN sc (x) = N 1/3 . −x

) −→ ({Ξ(t)}t∈[0,∞) , PAi),

in the vague topology.

(3.16)

N → ∞, (3.17)

Proofs of Results Proof of (i) of Theorems 2.6 and 2.7

To obtain (i) of the theorems it is enough to show µsin(Y) = 1 and µJν (Y+ ) = 1, since the (ν) other claims are derived from the facts that the operator Tt and Tt are of contraction and the set of all bounded continuous functions on M is dense in L2 (M, µsin) and L2 (M, µJν ). First we consider the probability measure µsin . Since its density ρsin is constant, the condition (3.3) is satisfied for ρ = ρsin . Note that any correlation ρm is bounded, because the kernel Ksin is bounded. Then if we take κ ∈ (1/2, 1) and m ∈ N satisfying (1 − κ)m > 1, 18

we see that the condition (3.8) is satisfied for ρ = ρsin . From Lemma 3.1 with k = 2, we see that the conditions (3.6) and (3.7) are satisfied with m′ R= 4, p = 2 and ρ = ρsin . Thus, ∞ µsin (Y) = 1. For µJν , its density ρ(ν) is bounded and satisfies 0 {ρ(ν) (x)/x}dx < ∞. Hence, by the same argument as above we obtain µJν (Y+ ) = 1. Remark 3. When µλ is the Poisson point process with an intensity measure λdx, λ > 0, ρm (xm ) = λm . Then we can readily confirm that all assumptions in Lemma 3.3 hold with m ∈ N, κ ∈ (1/2, 1) satisfying (1 − κ)m > 1 and with m′ = 4 and p = 2. Then µλ (Y) = 1. We can also show that measures such as Gibbs states with regular conditions are applicable to Lemma 3.3.

4.2

Proof of (ii) and (iii) of Theorems 2.6 and 2.7

The following lemma states stronger properties than (3.12) and (3.14) and is the key to proving (ii) and (iii) of Theorems 2.6 and 2.7. The proof of this lemma is given in the next subsection. Lemma 4.1 (i) For any 0 ≡ t0 < t1 < t2 < · · · < tM < ∞, M ∈ N0 , and bounded functions gj , 0 ≤ j ≤ M, which is Φ0 -moderately continuous on Ym,κ for any m ∈ N and κ ∈ (1/2, 1), # # "M "M Y Y gj (Ξ(tj )) . (4.1) gj (Ξ(tj )) = Esin lim EµGUE 2 N →∞

N,2N/π

j=0

j=0

f.d.

In particular, ({Ξ(t)}t∈[0,∞), PµGUE 2 ) −→ ({Ξ(t)}t∈[0,∞) , Psin), N → ∞, in the sense of N,2N/π finite dimensional distributions in the Φ0 -moderate topology. (ii) For any 0 ≡ t0 < t1 < t2 < · · · < tM < ∞, M ∈ N0 , and bounded functions gj , 0 ≤ j ≤ M, which is Φ0 -moderately continuous on Y+ m,κ for any m ∈ N and κ ∈ (1, 2), lim Eµ(ν)

N →∞

N,N

"M Y

#

gj (Ξ(tj )) = EJν

j=0

"M Y j=0

#

gj (Ξ(tj )) .

(4.2)

f.d.

In particular, ({Ξ(ν) (t)}t∈[0,∞) , Pµ(ν) ) −→ ({Ξ(t)}t∈[0,∞) , PJν ), N → ∞, in the sense of finite N,N dimensional distributions in the Φ0 -moderate topology. Proof of (ii) and (iii) in Theorems 2.6 and 2.7. We first give the proof for Theorem 2.6. Let fj , 0 ≤ j ≤ M be bounded vaguely continuous functions on M and 0 = t0 ≤ t1 < t2 < · · · < tM < ∞, M ∈ N. Theorem 2.6 (ii) is concluded from the equality "M # "M # Z Z Y Y µsin(dξ)Eξ fj (Ξ(tj )) = lim µGUE fj (Ξ(tj )) . (4.3) N,2N/π 2 (dξ)Eξ M

j=0

N →∞

19

M

j=0

Since Eξ

"M Y j=0

#

fj (Ξ(tj )) is Φ0 -moderately continuous on Ym,κ for any m ∈ N and κ ∈ (1/2, 1)

from Proposition 2.2 (ii), "M # (4.3) is guaranteed by (4.1) of Lemma 4.1 with M = 0 and Y g0 (ξ) = Eξ fj (Ξ(tj )) . Then Theorem 2.6 (ii) is proved. Now we show Theorem 2.6 (iii), j=0

which is derived from the relations: h i Esin f0 (Ξ(0))f1 (Ξ(t1 ))f2 (Ξ(t2 )) · · · fM (Ξ(tM ) h h = Esin f0 (Ξ(0))f1 (Ξ(t1 ))EΞ(t1 ) f2 (Ξ(t2 − t1 )) h i ii · · · EΞ(tM −1 −tM −2 ) fM (Ξ(tM − tM −1 )) · · · = hf0 , Tt1 (f1 Tt2 −t1 (f2 · · · TtM −tM −1 fM ) · · · )iµsin .

(4.4)

We show the first equality of (4.4) by induction with respect to M. First we consider the case M = 2. By the Markov property of ({Ξ(t)}t∈[0,∞) , PµGUE 2 ), we have N,2N/π

h i EµGUE 2 f0 (Ξ(0))f1 (Ξ(t1 ))f2 (Ξ(t2 )) N,2N/π h h ii = EµGUE 2 f0 (Ξ(0))f1 (Ξ(t1 ))EΞ(t1 ) f2 (Ξ(t2 − t1 )) N,2N/π

for 0 ≤ t1 < t2 < ∞. Since h i h i lim EµGUE 2 f0 (Ξ(0))f1 (Ξ(t1 ))f2 (Ξ(t2 )) = Esin f0 (Ξ(0))f1 (Ξ(t1 ))f2 (Ξ(t2 )) N →∞

N,2N/π

by (3.12), it is enough to show

h h ii lim EµGUE 2 f0 (Ξ(0))f1 (Ξ(t1 ))EΞ(t1 ) f2 (Ξ(t2 − t1 )) N,2N/π N →∞ h h ii = Esin f0 (Ξ(0))f1 (Ξ(t1 ))EΞ(t1 ) f2 (Ξ(t2 − t1 ))

(4.5)

h i for the proof. Since Eξ f2 (Ξ(t2 − t1 )) = Tt2 −t1 f2 (ξ) is Φ0 -moderately continuous on Ym,κ for any m ∈ N and κ ∈ (1/2, 1), (4.1) of Lemma 4.1 with M = 1 and g0 = f0 , g1 = f1 Tt2 −t1 f2 gives (4.5). Thus, we have obtained the case M = 2. Next we suppose that the first equality of (4.4) is satisfied in case M = k ∈ N. By using the same argument as in theQcase M = 2, k−1 fj , g1 = in which (4.1) of Lemma 4.1 is used in this case with M = 1 and g0 = j=0 fk Ttk+1 −tk fk+1 , we have Esin

"k+1 Y j=0

#

fj (Ξ(tj )) = Esin

"

k Y j=0

# h i fj (Ξ(tj ))EΞ(tk ) fk+1 (Ξ(tk+1 − tk )) 20

with t0 = 0. From the assumption of the induction the right hand side of the above equality equals to h h Esin f0 (Ξ(0))f1 (Ξ(t1 ))EΞ(t1 ) f2 (Ξ(t2 − t1 )) h i ii · · · EΞ(tk −tk−1 ) fk+1(Ξ(tk+1 − tk )) · · · .

Then we obtain the first equality of (4.4) in case M = k + 1, and the induction is completed. The second equality of (4.4) is derived from the definition (2.14) of the operator Tt and (2.15). That is, h h Esin f0 (Ξ(0))f1 (Ξ(t1 ))EΞ(t1 ) f2 (Ξ(t2 − t1 )) h i ii · · · EΞ(tM −1 −tM −2 ) fM (Ξ(tM − tM −1 )) · · · h i = Esin f0 (Ξ(0))f1 (Ξ(t1 ))Tt2 −t1 (f2 · · · TtM −tM −1 fM ) · · · )(Ξ(t1 )) = hf0 , Tt1 (f1 Tt2 −t1 (f2 · · · TtM −tM −1 fM ) · · · )iµsin .

The proof of Theorem 2.6 is completed. The proof of Theorem 2.7 is obtained by the same argument, in which (3.14) and (4.2) should be used instead of (3.12) and (4.1).

4.3

Proof of Lemma 4.1

We introduce subsets of Y,  γ,L0 Yκ,m = ξ ∈ M : m(ξ, κ) ≤ m,

Z

|x|≥L

 ξ(dx) −γ ≤ L , L ≥ L0 , x

with κ ∈ (1/2, 1) γ > 0, m, L0 ∈ N. Then we can prove the following lemma.   γ,L0 = 1 for some κ ∈ (1/2, 1) Lemma 4.2 (i) For any t > 0, lim lim min µGUE Y 2 κ,m N,2N/π +t m→∞ L0 →∞ N ∈N

and γ > 0.   (ν) (ii) For any t > 0, lim min µN,N +t Y+ κ,m = 1 for some κ ∈ (1, 2). m→∞ N ∈N

Proof. Here we give the proof of (i) only, since (ii) will be proved by the similar argument to the latter half of the proof of (i). For any fixed t > 0 we put ρN GUE (t, x)

 2 N −1 x 1 X ϕk √ , ≡ KN (t, x; t, x) = √ 2t k=0 2t

and N

ρ (t, x) ≡

ρN GUE



21

 2N + t, x , π2

(4.6)

Then ρN (t, x) is a symmetric function of x and bounded with respect to N and x. Since µGUE N,2N/π 2 +t is a determinantal point process, by Lemma 3.1 we have Z

M



µGUE N,2N/π 2 +t (dξ) ξ([0, L))

−

Z

0

L

4 ρ (t; x)dx ≤ CL2 N

with a positive constant C, which is independent of N. By Chebyshev’s inequality we have   Z L GUE N 7/8 µN,2N/π2 +t ξ([0, L)) − ≤ CL−3/2 , (4.7) ρ (t; x)dx ≥ L 0

and so

 Z GUE µN,2N/π2 +t ξ([0, L)) −

L 0

Similarly, we have µGUE N,2N/π 2 +t Note that

 Z ξ((−L, 0]) −

 −1/2 7/8 N ρ (t; x)dx ≤ L , ∀L ≥ L0 ≥ 1 − C ′ L0 .

0

 −1/2 7/8 ρ (t; x)dx ≤ L , ∀L ≥ L0 ≥ 1 − C ′ L0 . N

−L

Z

L≤|x|≤L′

ρN (t; x)dx = 0, x

L′ > L,

from the fact that ρN (t, x) is symmetric in x. Then, by the same procedure in the proof of Lemma 3.2 with C1 = 1 and ε = 7/8, we have   Z ξ(dx) −1/2 −1/8 GUE ≤ 24L , ∀L ≥ L0 ≥ 1 − 2C ′ L0 . (4.8) µN,2N/π2 +t x |x|≥L By Chebyshev’s inequality with Lemma 3.1, for any p ∈ N
p     3ρN (t, [g κ (k), g κ (k + 1)]) GUE κ κ µN,2N/π2 +t ξ [g (k), g (k + 1)] ≥ m ≤ |m − ρN (t, [g κ (k), g κ (k + 1)])|2p ≤ Ck (κ−1)p m−2p ,

(4.9)

R where ρN (t, D) ≡ D dxρN (t, x) and C is a constant independent of N and k. If p is large enough to satisfy (1 − κ)p > 1, we have     GUE κ κ µN,2N/π2 +t max ξ [g (k), g (k + 1)] ≥ m ≤ C ′ m−2p , k∈Z

and so we see lim min µGUE N,2N/π 2 +t m→∞ N ∈N



   κ κ max ξ [g (k), g (k + 1)] ≤ m = 1. k∈Z

22

(4.10)

Combining the above estimates (4.8) and (4.10), we obtain (i) of the lemma. Proof of Lemma 4.1. Since the proofs of (i) and (ii) are similar, here we only give the proof of (i). Let d(·, ·) be a metric on M associated with the vague topology. First remind that ξn converges Φ0 -moderately to ξ if the conditions (2.3) and (2.4) are satisfied. Then from the 0 definition of Yγ,L κ,m , we see that for any ε > 0 there exists δ > 0 such that |gj (ξ) − gj (η)| < ε,

0 ≤ j ≤ M,

0 for any ξ, η ∈ Yγ,L κ,m with d(ξ, η) < δ. Here we use the fact that the closure of {ξ ∈ M : m(ξ, κ) ≤ m} is a compact subset of M to ensure that δ does not depend on ξ or η. Then, from the fact (3.12), we can show that "M # "M # Y Y gj (Ξ(tj )) − Esin gj (Ξ(tj )) lim EµGUE 2 N →∞ N,2N/π j=0 j=0 ( ) M M Y X γ,L0 0 sup |gj (ξ)| (M + 1)µsin (M \ Yγ,L max µGUE ≤ κ,m ) + N,2N/π 2 +tj (M \ Yκ,m )

j=0

ξ

j=0

N ∈N

by using the Skorohod representation theorem, which can be applied to distributions on Polish spaces [1]. Hence, by Lemma 4.2 we obtain the lemma.

4.4

Proof of Theorem 2.8

We put N 1/3 ρN , 2N 2/3 + x), A (x) = ρGUE (N

(4.11)

where ρN GUE (t, x) is defined in (4.6). The soft-edge scaling limit (3.15) implies that lim ρN A (x) = ρAi (x) ≡ KAi (x, x).

N →∞

We also see the following estimates, whose proof will be given in Section 5. Lemma 4.3 There exists a positive constant C3 such that

and

In particular

|ρAi (x) − ρb(x)| ≤ |ρN bN A (x) − ρ sc (x)| ≤

C3 , |x|

C3 , |x|

x ∈ R,

(4.12)

x ∈ R, N ∈ N.

(4.13)

N ρA (x) − ρbN 2C3 sc (x) dx max ≤ L . N ∈N |x|≥L x Z

23

(4.14)

For the proof of (i) we first show µAi (YA ) = 1. Let ρAi be the density function of µAi . Note that ρAi (x) = O(|x|1/2 ), x → −∞ from (4.12). Using Lemma 3.1 with k = 3, we ′ see that (3.6) Qmand (3.7) are satisfied with m = 6, p = 4.5. Note the correlation inequality ρm (xm ) ≤ j=1 ρ(xj ) (see Proposition 4.3 in [32]). If we take κ ∈ (1/2, 2/3) and m ∈ N satisfying (3κ/2 − 1)m < −1, we see that the condition (3.8) is satisfied for the correlation functions ρm of µAi . The condition (3.4) with ρ = ρb is derived from (4.12) immediately. Hence, from Lemma 3.3 (iii) we obtain the desired result. For the proof of (ii) and (iii) we introduce subsets of YA , Z ) ( ξ(R) ρbsc (x)dx − ξ(dx) −γ A,γ,L0 Yκ,m = ξ ∈ M : m(ξ, κ) ≤ m, ≤ L , L ≥ L0 |x|≥L x

with κ ∈ (1/2, 2/3] γ > 0, m, L0 ∈ N. Then the following lemma is established. Lemma 4.4 We have lim lim min µGUE1/3 m→∞ L →∞ N ∈N N,N 0



0 τ2N 2/3 YA,γ,L κ,m



=1

for some κ ∈ (1/2, 2/3) and γ > 0. R∞ R0 N 3/2 Proof. Note that 0 ρN from Lemma 4.3. Since A (x)dx < ∞ and −L ρA (x)dx ≤ CL GUE µN,N 1/3 is a determinantal point process, by Lemma 3.1 Z

M



τ−2N 2/3 µGUE N,N 1/3 (dξ) ξ(−L, ∞))

−

Z

6 N ρA (x)dx

∞ −L

≤ CL9/2

with a positive constant C, which is independent of N. By Chebyshev’s inequality we have   Z ∞ C 23/24 N GUE ≤ 5/4 , ρA (x)dx ≥ L τ−2N 2/3 µN,N 1/3 ξ((−L, ∞)) − (4.15) L −L and so

τ−2N 2/3 µGUE N,N 1/3

 Z ξ((−L, ∞)) −

∞

−L



ρN A (x)dx

23/24

≤L

, ∀L ≥ L0



≥ 1−

C′ 1/4

L0

.

By using the estimate (4.13) and Lemma 3.2 (ii) with C1 = 1 and ε = 23/24, we see that   Z N C” ρ b (x)dx − ξ(dx) 2C ′ sc GUE (4.16) ≤ , ∀L ≥ L ≥ 1 − τ−2N 2/3 µN,N 1/3 0 L1/24 1/4 x L0 |x|≥L

with C ′′ = 72 + C3. Noting the estimate (4.13), by the same argument deriving (4.9) we will obtain     GUE κ κ τ−2N 2/3 µN,N 1/3 ξ [g (k), g (k + 1)] ≥ m ≤ Ck (3κ/2−1)p m−2p . 24

and lim min τ−2N 2/3 µGUE N,N 1/3 m→∞ N ∈N



   κ κ max ξ [g (k), g (k + 1)] ≤ m = 1. k∈Z

(4.17)

Combining the above estimates (4.16) and (4.17), we obtain the lemma. Proof of (ii) and (iii) of Theorem 2.8. For any fixed t > 0 we put s  s N 1/3 N  N 1/3 ρbN ρ b x , sc (t, x) = sc N 1/3 + t N 1/3 + t
N 1/3 1/2 1/3 1/2 ρN (t, x) = ρ N + t, 2N (N + t) + x , A GUE and

N ∈ N,

Z ) ξ(R) ρ b (t, x)dx − ξ(dx) sc 0 YA,γ,L (t) = ξ ∈ M : m(ξ, κ) ≤ m, ≤ L−γ , L ≥ L0 . κ,m |x|≥L x R Note that ρbN bN bN b(x), sc (t, ·) is a nonnegative function such that R dx ρ sc (t, x) = N, ρ sc (t, x) ր ρ N → ∞. By using the scaling property of the Dyson model, from Lemma 4.4 we have   A,γ,L0 τ Y (t) = 1, lim lim min µGUE 1/2 1/3 1/2 1/3 2N (N +t) κ,m N,N +t (

m→∞ L0 →∞ N ∈N

for some κ ∈ (1/2, 2/3) and γ > 0. Note that 2N 1/2 (N 1/3 + t)1/2 − 2N 2/3 + N 1/3 t − t2 /4 = O(N −1/3 ). Since Z N N ρ b (t, x)dx − ρ b (t, x + ε)dx sc sc ≤ εL−1/2 , x |x|≥L

we have

  A,γ,L0 lim lim min µGUE τ Y (t) = 1, 2/3 1/3 2 1/3 2N −tN +t /4 κ,m N,N +t

m→∞ L0 →∞ N ∈N

and so

lim lim min Pτ

m→∞ L0 →∞ N ∈N

µGUE1/3 −2N 2/3 N,N


0 ΞρbNsc (t) ∈ YA,γ,L (t) = 1 κ,m

(4.18)

for some κ ∈ (1/2, 2/3) and γ > 0. Let 0 ≡ t0 < t1 < t2 < · · · < tM < ∞, M ∈ N0 . Suppose that gj , 0 ≤ j ≤ M are bounded functions on M such that gj (ξ N ) → gj (ξ), N → ∞, if ξ ∈ YA and ξ N ∈ M with ξ N (R) = N N satisfy that ξ N → ξ vaguely, (2.9) holds with ρbN (·) = ρbN sc (tj , ·), and maxN ∈N m(ξ , κ) ≤ m for some m ∈ N and κ ∈ (1/2, 1). From (4.18), we can show "M # "M # Y Y gj (Ξ(tj )) = EAi lim Eτ 2/3 µGUE1/3 gj (Ξ(tj )) . N →∞

−2N

N,N

j=0

j=0

by the same argument as the proof of Lemma 4.1. Therefore, by applying (ii) of Proposition 2.5 we can show (ii) and (iii) of Theorem 2.8 by the same procedure as in the proof of Theorem 2.6. 25

5

Proof of Lemma 4.3

5.1

Proof of (4.12)

We use the following asymptotic expansions of the Airy functions in the classical sense of Poincar´e [37, 25, 26]: For x ≫ 1     2 3/2 2 3/2 x1/4 e− 3 x e− 3 x 2 3/2 2 3/2 ′ Ai(x) ≈ 1/2 1/4 L − x , Ai (x) ≈ − , M − x 2π x 3 2π 1/2 3          1 2 3/2 2 3/2 π 2 3/2 2 3/2 π Ai(−x) ≈ 1/2 1/4 sin Q + cos P , x − x x − x π x 3 4 3 3 4 3          x1/4 2 3/2 2 3/2 2 3/2 π 2 3/2 π ′ Ai (−x) ≈ 1/2 sin R S − cos , x − x x − x π 3 4 3 3 4 3 where L(z), M(z), P (z), Q(z), R(z) and S(z) are functions defined by ∞ 1 1 X vk 5 7 =1+ + O 2 , M(z) = =1− +O 2 , L(z) = zk 72z z zk 72z z k=0 k=0 ∞ ∞ 1 1 X X 5 k u2k+1 k u2k (−1) 2k+1 = +O 3 , P (z) = (−1) 2k = 1 + O 2 , Q(z) = z z z 72z z k=0 k=0 ∞ ∞ 1 1 X X v2k v2k+1 7 (−1)k 2k = 1 + O 2 , S(z) = R(z) = (−1)k 2k+1 = − +O 3 , z z z 72z z k=0 k=0 ∞ X uk

with uk =

and

6k + 1 Γ(3k + 1/2) , vk = − uk . Then k 54 k!Γ(k + 1/2) 6k − 1

 4 3/2  , ρAi (x) = (Ai′ (x))2 − x(Ai(x))2 = O e− 3 x

x → ∞,

ρAi (−x) = (Ai′ (−x))2 + x(Ai(−x))2       2  √ " x 7 2 3/2 π 1 2 3/2 π 1 sin 1+O 3 − − cos x − x − + O 9/2 = π 3 4 x 3 4 72x3/2 x      2 #    2 3/2 π 5 1 1 2 3/2 π + sin 1+O 3 x − + O 9/2 + cos x − 3/2 3 4 72x x 3 4 x      # √ " 2 3/2 π 2 3/2 π 1 x 1 1 + sin cos x − x − + O 3 , x → ∞. = 3/2 π 3 4 3 4 3x x Hence, we obtain

  ρAi (x) − ρb(x) = O 1 , |x| 26

x → ∞.

5.2

Proof of (4.13)

For proving (4.13) we use the asymptotic behavior of Hermite polynomials given in Plancherel b n (x) = (−1)n ex2 /2 (dn /dxn )e−x2 /2 , and Rotach [29], in which the Hermite polynomials are defined as H 2 2 whereas our definition√is Hn (x) = (−1)n ex (dn /dxn )e−x . We should note the relation that
b n (x) = 2−n/2 Hn x/ 2 . We introduce the following polynomials φn (z) and ψnp (z) deterH mined by the expansions ∞ ∞ h X (−1)m m−2 i X = φn (z)τ n , τ exp z m n=0 m=3 !p ∞ ∞ ∞ i X h X 1 X1 k−1 m−3 τ τ = ψnp (z)τ n , exp z k m n=0 m=4 k=1 for |τ | < 1. For example, φ0 (z) = 1, φ1 (z) = −z/3, φ2 (z) = z/4 + z 2 /18, φ3 (z) = −z/5 − z 2 /12 − z 3 /162, ψ0p (z) = 1, ψ1p (z) = p/2 + z/4. We set the coefficients of these polynomials P P (p) as φn (z) = nm=0 anm z m and ψnp (z) = nm=0 bnm z m . For example, 1 1 1 a00 = 1, a10 = 0, a11 = − , a20 = 0, a21 = , a22 = , 3 4 18 1 1 p (p) 1 1 (p) (p) , b00 = 1, b10 = , b11 = . a30 = 0, a31 = − , a32 = − , a33 = − 5 12 162 2 4 Then the asymptotic behaviors of Hermite polynomials given in [29] are summarized as follows. p (A.1) When x2 < 2(N + 1), we put x = 2(N + 1) cos θ, (0 < θ ≤ π/2). Then for any L∈N  2N/2 exp (N + 1)(1/2 + cos2 θ) HN (x) = N! (N + 1)(N +1)/2 (π sin θ)1/2 "L−1 n #   XX
N +1 1 1 3 −L/2 × Cnm (N, θ) sin , (2θ − sin 2θ) + Dnm (θ) + O (N sin θ) 2 n=0 m=0 where

) Γ(n + n+1 1 + (−1)n 2 anm , n/2 2 (N + 1) (sin θ)m+n/2 and   π θ π θ 1 . + Dnm (θ) = − − (2m + n) 4 2 4 2 p (A.2) When x2 > 2(N + 1), we put x = 2(N + 1) cosh θ, (0 < θ < ∞). Then for any L∈N  2N/2 exp (N + 1)(1/2 + cosh θ(cosh θ − sinh θ) HN (x) = N! (N + 1)(N +1)/2 (2π sinh θ)1/2 (cosh θ − sinh θ)(N +1/2) "L−1 n # XX
2 × Cnm (θ, N) + O N −L/2 θ−3L/2 ) , 1 Cnm (N, θ) =

n=0 m=0

27

where

 m+n/2 ) −2 1 + (−1)n Γ(n + n+1 2 anm . = 2 (N + 1)n/2 1 − e−2θ p (A.3) When x2 ∼ 2(N + 1), we put x = 2(N + 1) − 2−1/2 N −1/6 y, y = o(N 2/3 ). Then there exists a positive constant h∗ such "∞ #   3x2 /4 X A (x) HN (x) e ∗ 2 p = y p + O x−1/3 e−h x √
N +2/3 N! p! π x/ 2 p=0 2 Cnm (θ, N)

with the function Ap having the following asymptotic expansions:     L−1 X n X p+n+1 p+n+1 (−1)m 3m+n/3 (p−2)/3 (p) Ap (x) = 3 + m sin π bnm √
2n/3 Γ 3 3 x/ 2 n=0 m=0
+O x−2L/3 . Applying the above to the function ϕN (x) defined in (3.10), we obtain the lemma.

Lemma 5.1 (i) For N ∈ N, θ ∈ [0, π/2) and L ∈ N p  1 + O(N −1 )  2 1/4 ϕN 2(N + 1) cos θ = √ N π sin θ "L−1 n #   XX
N + 1 1 1 × Cnm (N, θ) sin (2θ − sin 2θ) + Dnm (θ) + O (N sin3 θ)−L/2 . 2 n=0 m=0

(5.1)

(ii) For N ∈ N and θ ∈ [0, ∞)     1 + O(N −1 )  1 1/4 p N +1 2(N + 1) cosh θ = √ ϕN (2θ − sinh 2θ) exp 2 2π sinh θ 2N "L−1 n #   XX 2 × Cnm (θ, N) + O N −L/2 θ−3L/2 ) . (5.2) n=0 m=0

(iii) For N ∈ N and y with |y| = o(N 2/3 )   p y 2(N + 1) − √ ϕN 2N 1/6     p y 1/4 −1/12 −1 B y, 2(N + 1) − √ =2 N + O(N ) , 2N 1/6 where o  x −2/3 n c10 Ai′ (−y) + c11 y 2Ai(−y) B(y, x) ≡ Ai(−y) + 2 o  x −4/3 n c20 Ai′ (−y) + c21 y 2Ai(−y) + c22 y 3Ai′ (−y) + 2 with constants cnm , 0 ≤ m ≤ n ≤ 2, which do not depend on x and y. 28

(5.3)

Proof. Applying Stirling’s formula   N  √ N 1 −2 + O(N ) N! = 2πN 1 + e 12N

to the results (A.1) and (A.2), we obtain (5.1) and (5.2) by simple calculation. To obtain (5.3) we use (A.3) with L = 3 and the expansion     ∞ 1 X (p−2)/3 p+1 2(p + 1) (−y)p Ai(−y) = 3 Γ sin π π p=0 3 3 p!     p ∞ p+1 p+1 y 1 X (p−2)/3 3 Γ sin π = π p=0 3 3 p! given in (2.36) in [37].

We show the main estimate in this section. √ Lemma 5.2 (i) Let x = 2N cos θ with N ∈ N and θ ∈ (0, π/2). Suppose that N sin3 θ ≥ CN ε for some C > 0 and ε > 0. Then   N −1  2 X 1 N . ϕk (x) = ρsc (1, x) + O √ 2 N sin θ k=0 √ (ii) Let x = 2N cosh θ with N ∈ N and θ > 0. Suppose that N sinh3 θ ≥ N ε for some C > 0 and ε > 0. Then   N −1  2 X 1 . ϕk (x) = O √ N sinh2 θ k=0 (iii) Let x = 2N 2/3 − y with N ∈ N and |y| ≤ CN β for some C > 0 and β ∈ (0, 2/21)
1/3 ρN , x) = ρAi (−y) + O |y|−1 . GUE (N

Proof. For the proof of this lemma we use the Christoffel-Darboux formula N −1  2  2 p X ϕk (x) = N ϕN (x) − N(N + 1)ϕN +1 (x)ϕN −1 (x) k=0

= N

n 2 o
ϕN (x) − ϕN +1 (x)ϕN −1 (x) 1 + O(N −1 ) .

(5.4)

For proving (i) we first show that for ℓ ∈ {−1, 0, 1}  1 + O(N −1 )  2 1/4 √ ϕN +ℓ 2N cos θ = √ N π sin θ " L−1 n   XX N 1 1 Cnm (N − 1, θ) sin × (2θ − sin 2θ) + Dnm (θ) − (1 + ℓ)θ 2 n=0 m=0 #  1  . (5.5) +O N sin θ 29

Substituting N − 1 instead of N in (5.1) and taking L > 2/ε, we have (5.5) with ℓ = −1. For calculating ϕN +ℓ (x) with ℓ ∈ {0, 1}, we take ηℓ such that s 2N cos(θ + ηℓ ) = cos θ, 2(N + 1 + ℓ) and then ϕN +ℓ Since

s

√

2N cos θ



2N = 2(N + 1 + ℓ)

= ϕN +ℓ r

p

 2(N + 1 + ℓ) cos(θ + ηℓ ) .

1 1+ℓ =1− +O 1− N +1+ℓ 2N



1 N2



.

(5.6) 

and cos(θ + ηℓ ) = cos θ − ηℓ sin θ + O(ηℓ2 ), we have

(1 + ℓ) cos θ ηℓ = +O 2N sin θ

From (5.1) and (5.6), we obtain



1 2 N sin θ



=O



1 N sin θ

(5.7)

 1/4 1 + O(N −1 ) 2 ϕN +ℓ 2N cos θ = √ N π sin θ " L−1 n   XX
N 1 1 × Cnm (N + ℓ, θ + ηℓ ) sin 2(θ + ηℓ ) − sin 2(θ + ηℓ ) + Dnm (θ + ηℓ ) 2 n=0 m=0 #   (5.8) +O (N sin3 θ)−L/2 . √



By simple calculations with (5.7), we have o N Nn sin 2(θ + ηℓ ) − 2(θ + ηℓ ) = (sin 2θ − 2θ) − (1 + ℓ)θ + O 2 2



1 N sin θ



.

Then from (5.8) with L > 2/ε and the above estimate, we obtain (5.5) for ℓ ∈ {0, 1}. Substituting (5.5) to (5.4)C from the identity 2 sin A sin B − sin(A + θ) sin(B − θ) − sin(A − θ) sin(B + θ) = 2 sin2 θ cos(A − B) 1 with A = N(2θ − sin 2θ)/2 + Dnm (θ) − θ and B = N(2θ − sin 2θ)/2 + Dn1 ′ m′ (θ) − θ, 0 ≤ m ≤ n ≤ L − 1, 0 ≤ m′ ≤ n′ ≤ L − 1, we have √   N −1  2 X √ 1 2N ϕk ( 2N cos θ) sin θ + O √ . = π N sin2 θ k=0

30

Since sin θ = N −1  X

√

ϕk (x)

k=0

1 − cos2 θ =

2

p

1 − x2 /{2(N + 1)} =

1√ = 2N − x2 + O π



1 √ N sin2 θ



p

2(N + 1) − x2 /

=

ρN sc (1, x)

+O



p

2(N + 1),

1 √ N sin2 θ



.

(5.9)

Hence, the proof of (i) is complete. For proving (ii) we show that for ℓ ∈ {−1, 0, 1} ϕN +ℓ

√

 1/4 1 1 + O(N −1 ) 2N cosh θ = √ 2π sinh θ 2N    N +1+ℓ × exp (2θ − sinh 2θ) + (1 + ℓ)θ 2 # " L−1 n  cosh3 θ  XX 2 . × Cnm (θ, N + ℓ) + O N sinh θ n=0 m=0 

(5.10)

Substituting N − 1 instead of N in (5.2) and taking L > 2/ε, we have (5.10) with ℓ = −1. For ℓ ∈ {0, 1}, take ηℓ such that s 2N cosh(θ + ηℓ ) = cosh θ, 2(N + 1 + ℓ) and we have ϕN +1

√

 p  2N cosh θ = ϕN +1 2(N + 1 + ℓ) cosh(θ + ηℓ ) .

Since cosh(θ + ηℓ ) = cosh θ + ηℓ sinh θ + O(ηℓ2 ), we have     (1 + ℓ) cosh θ cosh θ 1 ηℓ = − =O . +O 2N sinh θ N 2 sinh θ N sinh θ Substituting the equality to (5.2)Cwe have ϕN +ℓ

√

 1/4 1 + O(N −1 ) 1 2N cosh θ = √ 2π sinh θ 2N   o N +1+ℓ n 2(θ + ηℓ ) − sinh 2(θ + ηℓ ) × exp 2 " L−1 n #   XX 2 × Cnm (θ + ηℓ , N + ℓ) + O N −L/2 θ−3L/2 ) . 

n=0 m=0

Hence if we take L ∈ N such that L > 2/ε, from the relation

 cosh3 θ  1+ℓ , 2(θ + ηℓ ) − sinh 2(θ + ηℓ ) = 2θ − sinh 2θ + sinh 2θ + O N N 2 sinh θ 31

we can conclude (5.10) with ℓ ∈ {0, 1}. Combining (5.10) with (5.4), we obtain N −1  X

ϕk

k=0

√

2N cosh θ

2

h i N 1 √ = exp (N + 1)(2θ − sinh 2θ) + 2θ × O 2π sinh θ 2N



cosh3 θ N sinh θ



.

Then (ii) is concluded by simple calculation. √ Finally, we show (iii). Put w = 2N − 2−1/2 N −1/6 y, From the Christoffel-Darboux formula (5.4) and (5.3)   √ y ϕN −1 2N − √ 2N 1/6  o n  √ y −1 1/4 −1/12 −1 1/6 + O(N ) =2 N B y(1 − N ) , 2N − √ 2N 1/6 n o = 21/4 N −1/12 B (y, w) + O(N −1 ) .

Noting the following two simple relations,   p √ 1+ℓ −1/6 2N − cN = 2(N + 1 + ℓ) − c + √ N −1/6 + O(N −3/2 ), ℓ ∈ {0, 1}, 1/3 2N √ √ √ and y/ 2 + (1 + ℓ)/{ 2N 1/3 } = {y + (1 + ℓ)N −1/3 }/ 2, we apply (5.3) to obtain       √ y ℓ+1 −1 −1/3 1/4 −1/12 2N − √ + O(N ) . ϕN +ℓ B y + (1 + ℓ)N ,w − √ =2 N 2N 1/6 2N

Hence ϕN (w)2 − ϕN +1 (w) ϕN −1 (w) "  # 2   √ −1/6 1 1 B y + N −1/3 , w − √ B(y, w) − B y + 2N −1/3 , w − 2 √ = 2N 2N 2N +O(N −7/6 ). Noting that  2   1 1 −1/3 −1/3 B y+N ,w − √ B(y, w) − B y + 2N , w − 2√ 2N 2N ) (   1/2  2 y ∂2 ∂ −2/3 1+O B(y, w) − B(y, w) 2 B(y, w) N = ∂y ∂y wN 1/6 32

and the asymptotic properties of B(y, x) − Ai(−y) and its derivatives given by

B(y, x) − Ai(−y) = O(x−2/3 y 7/4 + x−4/3 y 13/4 ) ∂k ∂ℓ (B(y, x) − Ai(−y)) = O(x−2/3−ℓ y 7/4+k/2 + x−4/3−ℓ y 13/4+k/2 ), k ℓ ∂y ∂x x, y → ∞, we obtain  2 N −1 X √ y 1 √ 2N − √ ϕk 2N 1/6 k=0 2N 1/6 ) ( 2  1/2  y ∂2 ∂ B(y, w) − B(y, w) 2 B(y, w) 1+O = ∂y ∂y wN 1/6 ) ( 2  1/2  y ∂2 ∂ 5/2 −2/3 Ai(−y) − Ai(−y) 2 Ai(−y) + O(y w ) 1+O = ∂y ∂y wN 1/6  2 ∂ ∂2 = Ai(−y) − Ai(−y) 2 Ai(−y) + O(y 5/2 N −1/3 ). ∂y ∂y √ 1/6 Hence we conclude that for x = 2N w = 2N 2/3 − y, |y| ≤ N β with some β ∈ (0, 2/21), 2 
∂2 ∂ N 1/3 Ai(−y) − Ai(−y) 2 Ai(−y) + O |y|−1 . ρGUE (N , x) = ∂y ∂y

This completes the proof.

Putting x = 2N 2/3 (cos θ − 1) ∈ (−∞, 0). we have −x ∼ N 2/3 θ2 . In case N sin3 θ ≥ N 3ε/2 for some ε > 0, we see from Lemma 5.2 (i) that  
1 N N −1 ρA (x) = ρbsc (x) + O = ρbN . sc (x) + O |x| 2/3 2 N θ

Similarly, Putting x = 2N 2/3 (cosh θ − 1) ∈ (0, ∞), we have x ∼ N 2/3 (cosh θ − 1). In case N sinh3 θ ≥ N 3ε/2 for some ε > 0, we see from Lemma 5.2 (ii) that  
1 N −1 N N = ρ b (x) + O |x| . ρA (x) = ρbsc (x) + O sc N 2/3 sinh2 θ In the case that |x| ≤ N ε for some ε ∈ (0, 2/21), we see from Lemma 5.2 (ii) that
−1 ρN A (x) = ρAi (x) + O |x|
= ρbN ρN b(x)| + |b ρ(x) − ρAi (x)| + O |x|−1 . sc (x) + |b sc (x) − ρ From (3.16) and (4.12) we obtain (4.13).

Acknowledgements M.K. is supported in part by the Grant-in-Aid for Scientific Research (C) (No.21540397) of Japan Society for the Promotion of Science. H.T. is supported in part by the Grant-in-Aid for Scientific Research (KIBAN-C, No.23540122) of Japan Society for the Promotion of Science.

33

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