Infinite Particle Systems associated with Airy kernel Hideki - PowerPoint PPT Presentation

Infinite Particle Systems associated with Airy kernel Hideki TANEMURA (Chiba University) joint work with Hirofumi Osada (Kyushu University) 10th Workshop on Stochastic Analysis on Large Scale Interacting Systems 7th December 2011 1

1. Introduction Dyson’s Brownian motion model [Dyson 62] is a one parame- ter family of the systems solving the following stochastic differential equation: ∫ t X j ( t ) = x j + B j ( t ) + β ∑ ds X j ( s ) − X k ( s ) , 1 ≤ j ≤ n 2 0 k :1 ≤ k ≤ n k � = j where B j ( t ) , j = 1 , 2 , . . . , n are independent one dimensional Brown- ian motions. In the system, interaction between any pair of particles is repulsive and its strength is proportional to the inverse of particle distance with proportional constant β/ 2 > 0. We consider the case that β = 2 and call the model in the special case the Dyson model . 2

The Dyson model is realized by the following three processes: (i) The process of eigenvalues of Hermitian matrix valued diffusion process in the Gaussian unitary ensemble (GUE). (ii) The system of one-dimensional Brownian motions conditioned never to collide with each other. (iii) The harmonic transform of the absorbing Brownian motion in a Weyle chamber of type A n − 1 : { } W n = x = ( x 1 , x 2 , · · · , x n ) : x 1 < x 2 < · · · < x n . with harmonic function given by the Vandermonde determinant: [ ] ∏ x j − 1 h n ( x ) = ( x k − x j ) = det . k 1 ≤ j,k ≤ n 1 ≤ j<k ≤ n 3

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 ( · ) = δ x j ( · ) j ∈ I with some sequence ( x j ) j ∈ I of R satisfying ♯ { j ∈ I : x j ∈ K } < ∞ , for any compact set K . The index set I is countable. M is a Polish space with the vague topology: we say ξ n converges to ξ vaguely, if ∫ ∫ lim R ϕ ( x ) ξ n ( dx ) = R ϕ ( x ) ξ ( dx ) n →∞ for any ϕ ∈ C 0 ( R ), where C 0 ( R ) is the set of all continuous real- valued functions with compact supports. 4

For the solution ( X j ( t ) , j = 1 , 2 , , . . . , n ) of ∫ t ∑ ds X j ( t ) = x j + B j ( t ) + X j ( s ) − X k ( s ) , 1 ≤ j ≤ n, 0 k :1 ≤ k ≤ n k � = j we put n ∑ ξ n ( t ) = t ∈ [0 , ∞ ) , δ X j ( t ) , j =1 which is an M -valued diffusion process starting from the configura- n ∑ δ x j . We denote the process by ( ξ n ( t ) , P ξ ) and call it the tion ξ = j =1 Dyson model with unlabeled particles. 5

The moment generating function of multitime distribution of a M - valued process ξ ( t ) is defined as  } { ∫ M ∑ Ψ t ( f ) = E  exp  R f m ( x ) ξ ( t m , dx ) m =1 for t = ( t 1 , t 2 , . . . , t M ) with 0 ≤ t 1 < t 2 < · · · < t M , and f = ( f 1 , f 2 , . . . , f M ) with f m ∈ C 0 ( R ) , 1 ≤ m ≤ M . Set χ m ( · ) = e f m ( · ) − 1 , 1 ≤ m ≤ M .  ) ( ∫ N m M   ∑ ∏ 1 ∏ Ψ t ( f ) = N m ! d x ( m ) x ( m ) χ m ∏ M N m i   m =1 R Nm N m ≥ 0 , m =1 i =1 1 ≤ m ≤ M ( ) t 1 , x (1) N 1 ; . . . ; t M , x ( M ) × ρ , N M ( ) t 1 , x (1) N 1 ; . . . ; t M , x ( M ) with the multitime correlation functions ρ . N M 6

A processs ξ ( t ) is said to be determinantal if the moment gener- ating function of the multitime distribution is given by a Fredholm determinant [ ] Ψ t [ f ] = Det δ st δ x ( y ) + K ( s, x ; t, y ) χ t ( y ) , ( s,t ) ∈ ( t 1 ,t 2 ,...,t M ) 2 , ( x,y ) ∈ R 2 In other words, the multitime correlation functions are represented as   ( ) t 1 , x (1) N 1 ; . . . ; t M , x ( M )  K ( t m , x ( m ) ; t n , x ( n )  , ρ = det ) N M j k 1 ≤ j ≤ N m , 1 ≤ k ≤ N n 1 ≤ m,n ≤ M x ( m ) N m = ( x ( m ) , . . . , x ( m ) N m ) ∈ R N m , 1 ≤ m ≤ M , 0 < t 1 < · · · < t M < ∞ , 1 ( N 1 , . . . , N M ) ∈ N M , M ∈ N . The function K is called the correlation kernel of the determinantal process. 7

The Dyson model ξ n ( t ) starting from the origin is the determinantal process with the correlation kernel K n :  ( ) ( ) ( t ) k/ 2 n − 1  1 ∑ x y   √ √ √ ϕ k ϕ k , if s ≤ t ,    s 2 s 2 s 2 t k =0 K n ( s, x ; t, y ) = ( ) ( ) ( t ) k/ 2 ∞  − 1 ∑ x y   √ √ √ ϕ k ϕ k , if s > t .    2 s s 2 s 2 t k = n where ϕ k ( x ) = {√ π 2 k k ! } − 1 / 2 H k ( x ) e − x 2 / 2 is the normalized orthogonal functions on R comprising the Hermite polynomials H k ( x ) 8

As n → ∞ , the process ξ n ( n + t ) converges [ Bulk scaling limit ] the infinite dimensional determinantal process ( ξ sin ( t ) , P ) whose cor- relation kernel K sin comprising trigonometrical functions:  ∫ 1 1  0 du e ( t − s ) u 2 / 2 cos( u ( x − y )) ,   if s < t,   π    sin( x − y ) K sin ( s, x ; t, y ) = π ( x − y ) , if s = t,    ∫ ∞  − 1  du e ( t − s ) u 2 / 2 cos( u ( x − y )) ,   if s > t.  π 1 The process is reversible process with reversible measure µ sin the determinanatal point process with the sine kernel K sin ( x, y ) = K sin (0 , x ; 0 , y ) . [Nagao-Forrester (1998)] Theorem [Katori-T:to appear in MPRF] The process ( ξ sin ( t ) , P ) is a continuos reversible Markov process. 9

[ Soft edge scaling limit ] As n → ∞ , the scaled process θ a ( n,t ) ξ n ( n 1 / 3 + t ) ≡ { X j ( n 1 / 3 + t ) − a ( n, t ) } n j =1 , with a ( n, t ) = 2 n 2 / 3 + n 1 / 3 t − t 2 / 4, converges to the infinite dimen- sional determinantal process ( ξ Ai ( t ) , P ) whose correlation kernel K Ai comprising the Airy function Ai( x ):  ∫ 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 sin the determinanatal 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, [Forrester-Nagao-Honner (1999)],[Pr¨ ahofer-Spohn (2002)] Theorem [Katori-T:to appear in MPRF] The process ( ξ Ai ( t ) , P ) is a continuos reversible Markov process. 10

2. Dirichlet forms 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 . We put D 0 = { f : f is local and smooth with compact support } . Put ξ ( K ) ∂ ˜ D [ f, g ]( ξ ) = 1 ∑ f ∂ ˜ g , f, g ∈ D 0 , 2 ∂x j ∂x j j =1 and for a probability measure µ on M we introduce the bilinear form ∫ E µ ( f, g ) = M D [ f, g ] dµ, f, g ∈ D 0 . 11

Let Φ be a free potential, Ψ be an interaction potential. For a given sequence { b r } of positive integers we introduce a Hamiltonian on S r = ( − b r , b r ): ∑ ∑ H r ( ξ ) = H Φ , Ψ ( ξ ) = Φ( x j ) + Ψ( x j , x k ) r x j ∈ S r x j ,x k ∈ S r ,j<k We put M m r = { ξ ∈ M : ξ ( S r ) = m } and µ m r = µ ( · ∩ M m r ) . Definition (quasi Gibbs measure) A probability measure µ is said to be a (Φ , Ψ)-quasi Gibbs measure if there exists an increasing sequence { b r } of positive integers and measures { µ m r,k } such that for each r, m ∈ N satisfying µ m r,k ≤ µ m k →∞ µ m r,k = µ m lim r,k +1 , k ∈ N , r , weekly and that for all r, m, k ∈ N and for µ m r,k -a.s. ξ ∈ M c − 1 e − H r ( ζ ) 1 M m r ( ζ )Λ( dζ ) ≤ µ m r ) ≤ ce − H r ( ζ ) 1 M m r,k ( π S r ∈ dζ | ξ S c r ( ζ )Λ( dζ ) Here Λ is the Poisson random measure with intensity measure dx . 12

(A.1) µ has a locally bounded correlation functions ρ ( x n ) , n ∈ N . (A.2) µ is a (Φ , Ψ)-quasi Gibbs measure. (A.3) There exist upper semicontinuous functions Φ 0 , Ψ 0 , and pos- itive constants C and C ′ such that for any x, y ∈ R C − 1 Φ 0 ( x ) ≤ Φ( x ) ≤ C Φ 0 ( x ) , C ′− 1 Ψ 0 ( x − y ) ≤ Ψ( x, y ) ≤ C ′ Ψ 0 ( x − y ) , Ψ 0 ( x ) = Ψ 0 ( − x ) Moreover, Φ 0 and Ψ 0 are locally bounded from below and { Ψ 0 ( x ) = ∞} is compact. 13

Theorem [Osada :arXiv:math.PR/0902.3561] Assume (A.1), (A.2) and (A.3). Then (1) ( E µ , D 0 , L 2 ( M , µ )) is closable, (2) its closure ( E µ , D µ , L 2 ( M , µ )) is a local quasi regular Dirichlet space, (3) there exists a µ -reversible diffusion process (Ξ( t ) , P ) associated with the Diriclet space. Corollary [Osada :arXiv:math.PR/0902.3561] The probability measure µ sin satisfies (A.1), (A.2) and (A.3) with Φ( x ) = 0 and Ψ( x ) = − 2 log | x − y | , and there exists a µ sin -reversible diffusion process (Ξ sin ( t ) , P ) associated with the Diriclet space. 14

Theorem 1 The probability measure µ Ai satisfies (A.1), (A.2) and (A.3) with Φ( x ) = 0 and Ψ( x ) = − 2 log | x − y | , and there exists a µ Ai -reversible diffusion process (Ξ Ai ( t ) , P ) associated with the Diriclet space. It is proved that the process ( ξ Ai ( t ) , P ) is associated Remark. with a Diriclet space ( E , D ) which is a closed extension of the pre- Dirichlet space ( E µ Ai , D 0 , L 2 ( M , µ Ai )). Then we see that D µ Ai ⊂ D . Our conjecture is the coinsidence of the above two Dirichlet spaces, i.e. D µ Ai = D . 15