Stochastic Differential Equations Related to Soft-Edge Scaling Limit

Stochastic Differential Equations Related to Soft-Edge Scaling Limit Hideki Tanemura Chiba univ. (Japan) joint work with Hirofumi Osada (Kyushu Unv.) 2012 March 29 Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 1 / 26

Introduction The Gaussian ensembles are introduced as Hermitian matrices with independent elements of Gaussian random variables, and joint distribution invariant under conjugation by appropriate unitary matrices. The ensemble are divided into classes according to the element being real, complex or real quaternion, and their invariance under conjugation by orthogonal matices (GOE), unitary matrices (GUE), unitary symplectic matrices (GSE), respectively. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 2 / 26

Introduction The distribution of eigenvalues of the ensembles with size n n are given by { } mβ n (dx n) = 1 x i x j β exp β n x i 2 dx n, Z 4 i<j on the configuration space of n particles: i=1 W n = {x n R n : x 1 < x 2 < < x n } where the GOE, the GUE, and the GSE correspond with β = 1, 2 and 4, respectively. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 3 / 26

Introduction (Bulk scaling limit) For the eigenvalues {λ n 1,..., λn n} { nλ n 1,..., nλ n n} µ sin,β, weakly as n. µ sin,β is a probability measure on the configuration space of unlabelled particles: { } M = ξ : ξ is a nonnegative integer valued Radon measures in R Any element ξ of M can be represented as: ξ( ) = j I δ xj ( ) with some sequence (x j ) j I of R satisfying {j I : x j K} <, for any compact set K. M is a Polish space with the vague topology. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 4 / 26

Introduction For β = 2 (GUE) µ sin,2 is the determinantal point process(dpp), in which any spatial correlation function ρ m is given by a determinant with the sine kernel K sin,2 (x, y) = K sin (x, y) sin{π(y x)}, x, y R. π(y x) The moment generating function is given by a Fredholm determinant { } [ ] exp f (x)ξ(dx) µ sin,2 (dξ) = Det δ x (y) + K sin (x, y)χ(y), M R (x,y) R 2 for f C c (R), where χ( ) = e f ( ) 1. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 5 / 26

Introduction For β = 1 (GOE) µ sin,1 is the quaternion DPP with kernels [ K sin (x, y) y K sin,1 (x, y) = K ] sin(x, y) 1 2 sign(x y) + x y K. sin,2(u, y)du K sin (x, y) Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 6 / 26

Introduction For β = 1 (GOE) µ sin,1 is the quaternion DPP with kernels [ K sin (x, y) y K sin,1 (x, y) = K ] sin(x, y) 1 2 sign(x y) + x y K. sin,2(u, y)du K sin (x, y) Here we use a natural identification between the 2 2 complex matrices and the quaternions: ( ) ( ) ( ) ( ) 1 0 i 0 0 1 0 i I = e 0 1 1 = e 0 i 2 = e 1 0 3 = i 0 Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 6 / 26

Introduction For β = 1 (GOE) µ sin,1 is the quaternion DPP with kernels [ K sin (x, y) y K sin,1 (x, y) = K ] sin(x, y) 1 2 sign(x y) + x y K. sin,2(u, y)du K sin (x, y) Here we use a natural identification between the 2 2 complex matrices and the quaternions: ( ) ( ) ( ) ( ) 1 0 i 0 0 1 0 i I = e 0 1 1 = e 0 i 2 = e 1 0 3 = i 0 For β = 4 (GSE) µ sin,4 is the quaternion DPP with kernels [ K sin (2x, 2y) y K sin,4 (x, y) = K ] sin(2x, 2y) x y K. sin(2u, 2y)du K sin (2x, 2y) Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 6 / 26

Introduction The moment generating function is given by { } exp f (x)ξ(dx) µ sin,β (dξ) M R [ = Det δ x (y) I 2 + K sin,β (x, y)χ 2 (y) (x,y) R 2 [ ] = Pf δ x (y) I 2 + K sin,β (x, y)χ 2 (y), (x,y) R 2 ] 1/2 for f C c (R), where χ 2 ( ) = ( e f ( ) 1 e f ( ) 1 ), and Pf denotes the Fredholm pfaffian. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 7 / 26

Introduction n n Hermitian matrix valued process (n N) M(t) = M 11 (t) M 12 (t) M 1n (t) M 21 (t) M 22 (t) M 2n (t) M 1 (t) M n2 (t) M nn (t), M lk(t) = M kl (t). Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 8 / 26

Introduction n n Hermitian matrix valued process (n N) M(t) = M 11 (t) M 12 (t) M 1n (t) M 21 (t) M 22 (t) M 2n (t) M 1 (t) M n2 (t) M nn (t) (GOE) Bkl R (t), 1 k l n : indep. BMs, M lk(t) = M kl (t). M kl (t) = 1 Bkl R (t), 1 k < l n, 2 M kk (t) = Bkk R (t), 1 k n, Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 8 / 26

Introduction n n Hermitian matrix valued process (n N) M 11 (t) M 12 (t) M 1n (t) M(t) = M 21 (t) M 22 (t) M 2n (t), M lk(t) = M kl (t). M 1 (t) M n2 (t) M nn (t) (GOE) Bkl R (t), 1 k l n : indep. BMs M kl (t) = 1 Bkl R (t), 1 k < l n, 2 M kk (t) = Bkk R (t), 1 k n, (GUE) Bkl R (t), BI kl (t),1 k l n : indep. BMs M kl (t) = 1 1 Bkl R (t) + Bkl I (t), 1 k < l n, 2 2 M kk (t) = Bkk R (t), 1 k n, Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 8 / 26

Introduction (GSE) Bkl α (t), α = 0, 1, 2, 3, 1 k l n : indep. BMs For α = 1, 2, 3, Mkl 0 (t) = 1 Bkl 0 (t), 1 k < l n, 2 Mkk 0 (t) = B0 kk (t), 1 k n, 1 Mkl α (t) = Bkl α (t), 1 k < l n, 2 Mkk α (t) = 0, 1 k n, 2n 2n self dual Hermitian matrix valued process 3 M(t) = M 0 (t) I + M α (t) e α α=1 Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 9 / 26

Introduction The eigen-valued process is Dyson s Brownian motion model [Dyson 62], which is a one parameter family of the systems solving the following stochastic differential equation: dx j (t) = db j (t) + β 2 k:1 k n k j dt X j (s) X k (s), 1 j n (1) where B j (t), j = 1, 2,..., n are independent one dimensional Brownian motions. Osada[PTRF: online first] constructed the diffusion process which solves the SDE (1) with n = by the Dirichlet form technique associated with the measure µ sin,β, β = 1, 2 and 4. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 10 / 26

Introduction (Soft edge scaling limit) As n {n 1/6 λ n 1 2n 2/3,..., n 1/6 λ n n 2n 2/3 } µ Ai,β, weakly, For β = 2, µ Ai,2 is the determinantal point process with the Airy kernel Ai(x)Ai (y) Ai (x)ai(y) if x y K Ai (x, y) = x y (Ai (x)) 2 x(ai(x)) 2 if x = y, where Ai( ) is the Airy function defined by Ai(z) = 1 dk e i(zk+k3 /3), z C, 2π and Ai (x) = dai(x)/dx. R Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 11 / 26

Introduction For β = 1, µ Ai,1 is the quaternion DPP with the kernel [ K Ai (x, y) y K Ai,1 (x, y) = K ] Ai(x, y) 1 2 sign(x y) + x y K Ai(u, y)du K Ai (x, y) ( + 1 Ai(x) 1 ) Ai(u)du Ai(x)Ai(y) y ( 2 x y Ai(u)du 1 ) (1 Ai(u)du y Ai(u)du ). Ai(y) x For β = 4, µ Ai,4 is the quaternion DPP with the kernel [ K Ai,4 (x, y) = 1 [ + 1 2 2/3 2 1/3 K Ai (2 2/3 x, 2 2/3 y) y K Ai(2 2/3 x, 2 2/3 y) x y K Ai(2 2/3 u, 2 2/3 y)du K Ai (2 2/3 x, 2 2/3 y) Ai(2 2/3 x) ] Ai(2 2/3 u)du Ai(2 2/3 x)ai(2 2/3 y) y x y Ai(22/3 u)du Ai(2 2/3 u)du Ai(2 2/3 u)duai(2 2/3. y) y x ] Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 12 / 26

Results Let β = 1, 2 and 4. (1) There exists a diffusion process describing an infinite particle system, which has µ Ai,β as a reversible measure. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 13 / 26

Results Let β = 1, 2 and 4. (1) There exists a diffusion process describing an infinite particle system, which has µ Ai,β as a reversible measure. (2) The system solves the ISDE given by { dx j (t) = db j (t) + β 2 lim L k j, X k (t) <L 1 X j (t) X k (t) x <L j N, } ρ(x)dx dt, x where ρ(x) = x 1(x < 0). π Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 13 / 26

Dirichlet spaces A function f defined on the configuration space M is local if f (ξ) = f (ξ K ) for some compact set K. A local function f is smooth if f ( n j=1 δ x j ) = f (x 1, x 2,..., x n ) with some smooth function f on R n with compact support. Put D 0 = {f : f is local and smooth}. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 14 / 26

Dirichlet spaces A function f defined on the configuration space M is local if f (ξ) = f (ξ K ) for some compact set K. A local function f is smooth if f ( n j=1 δ x j ) = f (x 1, x 2,..., x n ) with some smooth function f on R n with compact support. Put We put D 0 = {f : f is local and smooth}. D[f, g](ξ) = 1 2 ξ(k) j=1 f (x) g(x) x j x j and for a probability measure µ we introduce the bilinear form E µ (f, g) = D[f, g]dµ, f, g D 0. M Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 14 / 26

quasi Gibbs measure Let Φ be a free potential, Ψ be an interaction potential. For a given sequence {b r } of N we introduce a Hamiltonian on I r = ( b r, b r ): H r (ξ) = H Φ,Ψ r (ξ) = x j I r Φ(x j ) + x j,x k I r,j<k Ψ(x j, x k ) Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 15 / 26

quasi Gibbs measure Let Φ be a free potential, Ψ be an interaction potential. For a given sequence {b r } of N we introduce a Hamiltonian on I r = ( b r, b r ): H r (ξ) = Hr Φ,Ψ (ξ) = Φ(x j ) + Ψ(x j, x k ) x j I r x j,x k I r,j<k Definition A probability measure µ is said to be a (Φ, Ψ)-quasi Gibbs measure if there exists an increasing sequence {b r } of N and measures {µ m r,k } such that for each r, m N satisfying µ m r,k µm r,k+1, k N, lim k µm r,k = µ( {ξ(i r ) = m}), weekly and that for all r, m, k N and for µ m r,k -a.s. ξ M c 1 e H r (ξ) 1 {ξ(ir )=m}λ(dζ) µ m r,k (π I r dζ ξ I c r ) ce H r (ξ) 1 {ξ(ir )=m}λ(dζ) Here Λ is the Poisson random measure with intensity measure dx. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 15 / 26

quasi regular Dirichlet space Theorem (Bulk) [Osada:to appear in AOP] Let β = 1, 2, 4. (1) The probability measure µ sin,β is a quasi Gibbs measure with Φ(x) = 0 and Ψ(x) = β log x y. (2) The closure of (E µ sin,β, D 0, L 2 (M, µ sin,β )) is a quasi Dirichlet space, and there exists a µ sin -reversible diffusion process (Ξ sin,β (t), P) associated with the Diriclet space. Theorem 1 (Soft-Edge) [Osada-T] Let β = 1, 2, 4. (1) The probability measure µ Ai,β is a quasi Gibbs measure with Φ(x) = 0 and Ψ(x) = β log x y. (2) The closure of (E µ Ai,β, D 0, L 2 (M, µ Ai,β )) is a quasi Dirichlet space, and there exists a µ Ai -reversible diffusion process (Ξ Ai,β (t), P) associated with the Diriclet space. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 16 / 26

Extended Airy kerenel Let (ξ Ai (t), P) be the determinantal process with the extended Airy kernel 0 du e (t s)u/2 Ai(x u)ai(y u), if s t, K Ai (s, x; t, y) = du e (t s)u/2 Ai(x u)ai(y u), if s > t. 0 The process is reversible process with reversible measure µ Ai,2. [Forrester-Nagao-Honner (1999)],[Prähofer-Spohn (2002)] Theorem [Katori-T: MPRF(2011)] The process (ξ Ai (t), P) is a continuos reversible Markov process. Open problem. (ξ Ai (t), P) = (Ξ Ai,2 (t), P)? Proposition [Katori-T: JSP(2007)] The Dirichlet space of the process (ξ Ai (t), P) is a closed extension of (E µ Ai, D 0, L 2 (M, µ Ai )). Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 17 / 26

Infinite dimensional Stochstic Differential Equations Theorem 2[Osada-T] Let β = 1, 2, 4. The process Ξ Ai,β (t) = j N δ X j (t) satisfies dx j (t) = db j (t) + β { 2 lim L k j: X k (t) <L 1 X j (t) X k (t) } ρ(x) x dx dt, j N, x <L where B j (t), j N are independent Brownian motions and ρ(x) = x 1(x < 0). π Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 18 / 26

Log derivative Let µ x be the Palm measure conditioned at x = (x 1,..., x k R k ( µ x = µ k j=1 ) δ xj ξ(x j ) 1 for j = 1, 2,..., k. Let µ k be the Campbell measure of µ: µ k (A B) = µ x (B)ρ k (x)dx, A B(R k ), B B(M). A We call d µ L 1 loc (R M, µ1 ) the log derivative of µ if d µ satisfies d µ (x, η)f (x, η)dµ 1 (x, η) = x f (x, η)dµ 1 (x, η), R M R M f C c (R) D 0. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 19 / 26

ISDE Theorem [Osada, PTRF (on line first)] Assume that there exists a log derivative d µ (and some conditions). There exists M 0 M such that µ(m 0 ) = 1, and for any ξ = j N δ x j M 0, there exists R N -valued continuous process X(t) = (X j (t)) j=1 satisfying X(0) = x = (x j ) j=1 and dx j (t) = db j (t) + 1 2 dµ ( X j (t), k:k j δ Xk (t) ) dt, j N. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 20 / 26

ISDE Theorem [Osada, PTRF (on line first)] Assume that there exists a log derivative d µ (and some conditions). There exists M 0 M such that µ(m 0 ) = 1, and for any ξ = j N δ x j M 0, there exists R N -valued continuous process X(t) = (X j (t)) j=1 satisfying X(0) = x = (x j ) j=1 and dx j (t) = db j (t) + 1 ( 2 dµ X j (t), ) dt, j N. k:k j δ Xk (t) Lemma (Bulk) [Osada, PTRF online first] η = δ yj with η({x}) = 0, j N Let β = 1, 2, 4. For x R and d µ sin,β (x, η) = β lim 1 x y j L j: x y j L Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 20 / 26

Key lemma The key part in the proof of Theorem 2 is to determine the log derivative of µ Ai,β. Key lemma (Soft-Edge) Let β = 1, 2, 4. For x R and η = δ yj j N with η({x}) = 0, d µ Ai,β (x, η) = β lim L j: x y j L 1 ρ(u) x y j u L u du. Lemma (Bulk) [Osada, PTRF online first] η = δ yj with η({x}) = 0, j N Let β = 1, 2, 4. For x R and d µ sin,β (x, η) = β lim L j: x y j L 1 x y j u L ρ u du. Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 21 / 26

Proof To prove the key lemma, we use n particle sysytem: mβ n (du n) = 1 { u i u j β exp β n u i }du 2 n, Z 4 i<j We put u j = 2 n + x j and intrduce the measure defined by n 1/6 { µ n A,β (dx n) = 1 x i x j β exp β n 2 } n + n 1/6 x i 2 dx n, Z 4 i<j The log derivative d n of the measure µ n A,β is given by n 1 { n 1 } d n (x, η) = d n x, δ yj 1 = β n 1/3 n 1/3 x. x y j 2 j=1 j=1 i=1 i=1 Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 22 / 26

Proof Lemma 3 is derived from the fact that d µai (x, η) = lim n dn (x, η) = β lim L x y j <L 1 ρ(y) x y j u L y du (2) Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 23 / 26

Proof Lemma 3 is derived from the fact that d µai (x, η) = lim n dn (x, η) = β lim L x y j <L To check (2) we divide d n /β into three parts: g n L (x, η) = 1 x y j wl n (x, η) = u n (x) = R x y j <L x y j L ρ n A,β,x (u) x u 1 x y j 1 ρ(y) x y j u L y du x u <L x u L du n1/3 n 1/3 x. 2 ρ n A,β,x (u) x u du, ρ n A,β,x (u) x u du, Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 23 / 26 (2)

Proof The fact (2) is obtained if the following conditions hold: with and u(x) = lim L lim n gn L (x, η) = g L(x, η), in Lˆp (µ 1 Ai,β ) for any L > 0, (3) lim lim sup wl n (x, y) ˆp dµ n,1 L A (dxdη) = 0, (4) n [ r,r] M lim n un (x) = u(x), in Lˆp loc (R, dx), (5) g L (x, η) = { u L x y j <L 1 x y j x u <L ρ Ai,β,x (u) x u du, } ρ Ai,β,x (u) x u du ρ(u) u L u du Lˆp loc (R, dx). Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 24 / 26

Proof In the case β = 2, µ n A is the DPP with the correlation kernel KA(x, n y) = n 1/3 Ψ n(x)ψ n 1 (y) Ψ n 1 (x)ψ n (y) x y ( 2n ) where Ψ n (x) = n 1/12 φ n + x 2n 1/6, and φ k (x) is the normalized orthogonal functions on R comprising the Hermite polynomials H k (x). Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 25 / 26

Proof In the case β = 2, µ n A is the DPP with the correlation kernel KA(x, n y) = n 1/3 Ψ n(x)ψ n 1 (y) Ψ n 1 (x)ψ n (y) x y ( 2n ) where Ψ n (x) = n 1/12 φ n + x 2n 1/6, and φ k (x) is the normalized orthogonal functions on R comprising the Hermite polynomials H k (x). We also use the function ρ n (u) = 1 ( u 1 + u ) π 4n 2/3, 4n 2/3 u 0 and the facts R ρ n (u) u du = n1/3, lim n ρn (u) ρ(u). Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 25 / 26

Thanks Thank you for your attention! Hideki Tanemura (Chiba univ.) () SDEs related to Soft-Edge scalng limit 2012 March 29 26 / 26