Zeros of Airy Function and Relaxation Process

Zeros of Airy F unction and Relaxation Pro cess Mak oto Kator i ∗ and Hidek i T a nem ura † 29 September 2 009 Abstract One-dimensional system of Bro wnian motions called Dyson’s m o del is the particle system with long-range repu lsiv e forces acting b et wee n any pair of particles, where the strength of force is β / 2 times the in v erse of p artic le distance. When β = 2, it is realized as the Bro wn ian motions in one di- mension cond itioned nev er to collide with eac h other. F or an y initial con- figuration, it is pr o v ed that Dyson’s mo del with β = 2 and N particles, X ( t ) = ( X 1 ( t ) , . . . , X N ( t )) , t ∈ [0 , ∞ ) , 2 ≤ N < ∞ , is d et erminant al in the sense that an y multitime correlati on fu ncti on is giv en by a determinan t with a con tinuous ke rnel. The Airy fun ction Ai( z ) is an entire fu nctio n with zeros all lo cate d on the negativ e part of the real axis R . W e con- sider Dyson’s mo del with β = 2 starting from the first N zeros of Ai( z ), 0 > a 1 > · · · > a N , N ≥ 2. In order to pr o p erly cont rol the effect of suc h ini- tial confinemen t of particle s in the negativ e r egion of R , we pu t the drift term to eac h Bro wnian m o tion, whic h increases in time as a parab olic f unction : Y j ( t ) = X j ( t )+ t 2 / 4+ { d 1 + P N ℓ =1 (1 /a ℓ ) } t, 1 ≤ j ≤ N , where d 1 = Ai ′ (0) / Ai(0). W e show that, as the N → ∞ limit of Y ( t ) = ( Y 1 ( t ) , . . . , Y N ( t )) , t ∈ [0 , ∞ ), w e obtain an in finite particle system, wh ic h is the r e laxation pro cess from the configuration, in whic h ev ery zero of Ai( z ) on the negativ e R is o cc upied by one particle, to the s tationary state µ Ai . T he stationary state µ Ai is the deter- minan tal p oin t process with th e Airy k ernel, whic h is spatia lly inh o mogeneous on R and in whic h th e T racy-Widom distribution describ es the righ tmost par- ticle p osition. KEY W O R DS: Zeros of Airy fu ncti on; Relaxation process; Dyson’s mo del; Determinan tal p oin t p rocess; Entire f unction; W eierstrass canonical pro duct ∗ Department of Ph ys ics, F acult y of Science and Engineering, Chuo Universit y , Kasuga , Bunkyo- ku, T oky o 112-855 1, J apan; e-mail: k atori@phys.ch uo-u.ac .jp † Department o f Mathematics and Infor matics, F acult y of Science, Chiba Universit y , 1-3 3 Y ay oi- cho, Ina ge-ku, Chiba 263 -8522, Japa n; e- mail: tanemura@math.s.chiba-u.ac.jp 1 1 In tro d uction 1.1 Dyson’s mo del: On e-Di mensional B ro wnian P article Sys- tem In teracting through P air F orce 1 /x T o understand the time-evolution o f distributions of interacting particle systems on a large space-time scale ( thermo dynamic and hydr o dynamic limits ) is one o f the main topics o f statistical ph ysics. If the interactions a mong particles are short ranged, the standard theory is useful. If they are long r a nged, ho we v er, general theory has not y et b een established and thus detailed study of mo de l system s is required [26]. In the prese n t pap er, w e consider Bro wnian particles in one dimension with long- ranged repulsiv e forces acting b et w een any pair of particles, where the strength of force is exactly equal to 1 /x when the particle distance is x . If the n umber of particles is finite N < ∞ , the system is describ ed b y Ξ( t ) = P N j =1 δ X j ( t ) , 2 ≤ N < ∞ , where X ( t ) = ( X 1 ( t ) , . . . , X N ( t )) satisfies the follow ing system of sto c hastic differential equations (SDEs); dX j ( t ) = dB j ( t ) + X 1 ≤ k ≤ N k 6 = j 1 X j ( t ) − X k ( t ) dt, 1 ≤ j ≤ N , t ∈ [0 , ∞ ) (1.1) with indep enden t one-dimensional standard Brownian motions B j ( t ) , 1 ≤ j ≤ N . The SDEs obta ine d b y replacing the 1 /x force in (1.1) b y β / (2 x ) with a parameter β > 0 we re introduced by Dyson [4] to understand the statistics of eigen v alues o f hermitian r andom matric es as pa r t icle distributions of in teracting Brow nian mo- tions in R . Corresp onding to the special v alues β = 1 , 2 and 4, hermitian random matrices are in the three statistical ensem bles with differen t symmetries, called the Gaussian orthog onal e nsem ble (GOE), the G a us sian unitary ensem ble (G UE ), and the Gaussian symplectic ensem ble (GSE), resp ec tiv ely [5]. In particular for β = 2, that is the case of (1.1), if the eigen v alue distribution of N × N hermitian ra ndom matrices in the G UE with v ariance σ 2 is denoted b y µ GUE N ,σ 2 , w e can sho w lim N →∞ µ GUE N , 2 N/π 2 ( · ) = µ sin ( · ) , (1.2) where µ sin denotes the determinan t al (F ermion) p oint pr o c ess [2 4, 23] with the so- called sine kernel K sin ( x ) = 1 2 π Z | k |≤ π dk e √ − 1 k x = sin( π x ) π x , x ∈ R . (1.3) That is, µ sin is a spatially homogeneous particle distribution, in whic h the particle densit y is g iven b y ρ sin = lim x → 0 K sin ( x ) = 1 and an y N 1 -p oin t correlation function ρ sin ( x N 1 ) , x N 1 = ( x 1 , . . . , x N 1 ) ∈ R N 1 , N 1 ≥ 2 , is g iven by a determinan t of a n N 1 × N 1 real symme tric matrix; ρ sin ( x N 1 ) = det 1 ≤ j,k ≤ N 1 h K sin ( x j − x k ) i . 2 Based on this fact known in the ra ndom matrix theory [17], Sp ohn [25] studied the equilibrium dynamics obtained in the infinite-particle limit N → ∞ of Dyson’s mo del (1.1) . Since the 1 /x force is not summable, in the infinite-pa r ticle limit N → ∞ the sum in (1.1) should be regarded as an improp er s um, in the se nse that for X j ( t ) ∈ [ − L, L ] the summation is restricted to k ’s suc h that X k ( t ) ∈ [ − L, L ] and then t he limit L → ∞ is tak en. It is exp ected that the dynamics with a n infinite num b er of particles can exist only for in itial c onfigurations h a ving the same asymptotic densit y to the righ t and left [25, 1 3]. The problem, whic h we address in the presen t pap er, is how w e can con trol Dyson’s mo del with an infinite n umber of particles starting from asymm etric initial configurations. The motiv ation is ag ain coming from the ra ndom matrix theory as follo ws. Consider the A iry function [1, 30] Ai( z ) = 1 2 π Z R dk e √ − 1( z k + k 3 / 3) . (1.4) It is a solution o f Airy’s equation f ′′ ( z ) − z f ( z ) = 0 with the asymptotics on the real axis R : Ai( x ) ≃ 1 2 √ π x 1 / 4 exp  − 2 3 x 3 / 2  , Ai( − x ) ≃ 1 √ π x 1 / 4 cos  2 3 x 3 / 2 − π 4  in x → + ∞ . (1.5) In the GUE random matrix theory , the follo wing scaling limit has been extensiv ely studied: lim N →∞ µ GUE N ,N 1 / 3 (2 N 2 / 3 + · ) = µ Ai ( · ) , (1.6) where µ Ai is the determinan tal p oin t pro cess suc h that t he correlation k ernel is giv en b y [8, 28], K Ai ( y | x ) = Z ∞ 0 du Ai( u + x )Ai( u + y ) =    Ai( x )Ai ′ ( y ) − Ai ′ ( x )Ai( y ) x − y , x 6 = y ∈ R (Ai ′ ( x )) 2 − x (Ai( x )) 2 , x = y ∈ R . (1.7) It is another infinite-pa rticle limit different from (1.2) and is called the soft-e dge sc al- ing lim it , s ince x 2 / 2 t ≃ ( 2 N 2 / 3 ) 2 / (2 N 1 / 3 ) = 2 N marks the righ t edge of semicircle- shap ed profile of the GUE eigenv alue distribution (see, for example, [1 2 ]). The particle distribution µ Ai with the A iry kernel ( 1 .7 ) is highly asymmetric: As a mat- ter of fact, the particle densit y ρ Ai ( x ) = K Ai ( x | x ) de ca ys rapidly to zero as x → ∞ , but it div erges ρ Ai ( x ) ≃ 1 π ( − x ) 1 / 2 → ∞ as x → −∞ . (1.8) 3 Let R be the p osition of the rightmost particle on R in µ Ai . Then its dis tribution is giv en by the celebrated T r acy-Widom distribution [28] µ Ai ( R < x ) = exp  − Z ∞ x ( y − x )( q ( y )) 2 dy  , where q ( x ) is the unique solution of the P ainlev ´ e I I equation q ′′ = xq + 2 q 3 satisfying the b oundary condition q ( x ) ≃ Ai( x ) in x → ∞ . Pr¨ ahofer and Sp ohn [22] and Johansson [11 ] studied the equilibrium fluctuation of this r ig h tmost particle a nd called it the Airy pr o c ess . T racy and W idom derive d a system of partial differential equations, whic h go v ern the Airy pro cess [29]. See a lso [2 , 3]. How can we realize µ Ai as the equilibrium state of Bro wnian infinite-particle system interacting through pair force 1 /x ? The initial configura tions should b e a symmetric, but what kinds of conditions should b e satisfied by them ? How should w e mo dify the SD Es o f original D yson’s mo del (1.1), w hen w e provide finite-particle approximations for suc h asymmetric infinite particle systems ? In the presen t pap er, as an explicit answ er to the ab o v e ques tions, we will presen t a relaxatio n pro cess with an infinite n um b er of pa r t icles con v erging to the stationar y state µ Ai in t → ∞ . Its initial configuration is giv en b y ξ A ( · ) = X a ∈A δ a ( · ) = ∞ X j =1 δ a j ( · ) , (1.9) in whic h ev ery zero of the Airy function (1.4) is o ccupied b y one part icle. This sp ec ial choice of t he initial configura tion is due to the fact that the zeros o f the Airy function are locat ed only on the negative part of the real axis R , A ≡ Ai − 1 (0) = n a j , j ∈ N : Ai( a j ) = 0 , 0 > a 1 > a 2 > · · · o , (1.10) with the v alues [1] a 1 = − 2 . 33 . . . , a 2 = − 4 . 08 . . . , a 3 = − 5 . 52 . . . , a 4 = − 6 . 78 . . . , and that they a dm it the a s ymptotics [1, 30] a j ≃ −  3 π 2  2 / 3 j 2 / 3 in j → ∞ . (1.11) Then the av erage densit y of zeros of the Airy function around x , denoted by ρ Ai − 1 (0) ( x ), b eha v es as ρ Ai − 1 (0) ( x ) ≃ 1 π ( − x ) 1 / 2 → ∞ as x → −∞ , whic h coincides with (1.8). The appro ximation of our pro cess with a finite n um b er of particles N < ∞ is giv en b y Ξ A ( t ) = P N j =1 δ Y j ( t ) with Y j ( t ) = X j ( t ) + t 2 4 + D A N t, 1 ≤ j ≤ N , t ∈ [0 , ∞ ) , (1.12) 4 asso ciated with the solution X ( t ) = ( X 1 ( t ) , . . . , X N ( t )) o f Dyson’s mo del (1.1), where D A N = d 1 + N X ℓ =1 1 a ℓ . (1.13) Here d 1 = Ai ′ (0) / Ai(0) and A N ≡ n 0 > a 1 > · · · > a N o ⊂ A is the s equence of the first N zeros of the Airy function. In other w o r ds, Y ( t ) = ( Y 1 ( t ) , Y 2 ( t ) , . . . , Y N ( t )) satisfies the follo wing SDEs ; d Y j ( t ) = dB j ( t ) +  t 2 + D A N  dt + X 1 ≤ k ≤ N k 6 = j dt Y j ( t ) − Y k ( t ) = dB j ( t ) + X 1 ≤ k ≤ N k 6 = j  1 Y j ( t ) − Y k ( t ) + 1 a k  dt +  t 2 + d 1 + 1 a j  dt, 1 ≤ j ≤ N , t ∈ [0 , ∞ ) , (1.14) where B j ( t )’s are indep enden t one-dimensional standard Bro wnian motions. F or Y ( 0 ) = x ∈ R N , set ξ N ( · ) = P N j =1 δ x j ( · ) and consider the pro cess Ξ A ( t ) starting from the configurat io n ξ N . W e consider a set of initial configurations ξ N suc h that they are in general differen t from the N -pa r t ic le appro ximation of (1.9), ξ N A ( · ) = X a ∈A N δ a ( · ) = N X j =1 δ a j ( · ) , (1.15) but the particle densit y ρ ( x ) of lim N →∞ ξ N will show the same asymptotic in x → −∞ as (1.8). Because of the strong repulsiv e forces acting betw een particle pairs in (1.1), suc h confinemen t of par t icles in the negative region o f R at the initial time causes strong p ositive d rifts of Brow nian pa r ticle s. The co efficien t ( 1.13) of the drift term D A N t added in (1.1 2), how ev er, n e gativel y d iver ge s D A N ≃ −  12 π 2  1 / 3 N 1 / 3 → −∞ as N → ∞ . (1.16) W e will determine a class of asymmetric initial configur a tions denoted b y X A 0 , whic h includes ξ A as a t ypical one, suc h that the effect on dynamics of asymmetry in configuration will b e comp ensated b y the additional drift term D A N t in the infinite- particle limit N → ∞ and the dynamics with an infinite num b er of particles exists. Note that w e should take N → ∞ limit for finite t < ∞ in our process (1.1 2 ) to discuss non-equilibrium dynamics with an infinite n um b er of particles. In the class X A 0 , when the initial configura t io n is sp ecially set to be ξ A , (1.9), w e c an prov e that the dynamics sho ws a relaxation in the long-term limit t → ∞ to the equilibrium dynamics in µ Ai . In the pro of w e use t he sp ec ial prop ert y of the systems (1.1) 5 and (1 .1 4 ) suc h that the pro cesses hav e space-time determinan tal correlations. This feature comes from the f act that if and only if the strength of pa ir force is exactly equal to 1 /x when the particle distance is x , i.e. , iff β = 2, Dyson’s mo del is realized as the Bro wnian motions c onditione d never to c ol lide with e ac h other [10, 12]. In order to explain the imp ortance o f the notion of entir e functions for the presen t problem, w e rewrite the results reported in our previous paper [13] fo r D y son’s mo del with symmetric initial configurat io ns b elo w. Then the c ha ng e s whic h w e hav e to do for the systems with asymmetric initial configuratio ns are sho wn. There the origin of the quadra t ic term t 2 / 4 in (1.12) will b e clarified. 1.2 Pro cesses with Space-time Determinan tal Correlations and En tire F un ctio ns In an earlier pap er [13], w e studied a class of a non-equilibrium dynamics of Dyson’s mo del with β = 2 and an infinite n um b er of particles. As an example in the class, w e rep orted a r e laxation pro cess, denoted here by (Ξ( t ) , P sin ), whic h starts f r o m a configuration ξ Z ( · ) = X a ∈ Z δ a ( · ) , (1.17) in whic h ev ery p oin t of Z is o ccupied by one particle, and con v erges to the sta- tionary state µ sin . This pro cess (Ξ( t ) , P sin ) is determinantal , in the sense that there is a function K sin ( s, x ; t, y ) called the c orr elation kernel such that it is con- tin uo us with resp ect to ( x, y ) ∈ R 2 for an y fixed ( s, t ) ∈ [0 , ∞ ) 2 , and that, for an y integer M ≥ 1, an y seque nce ( N m ) M m =1 of p ositiv e integers, and any time se - quence 0 < t 1 < · · · < t M < ∞ , the ( N 1 , . . . , N M )- multitime c orr elation function ρ sin ( t 1 , x (1) N 1 ; . . . ; t M , x ( M ) N M ) , x ( m ) N m = ( x ( m ) 1 , . . . , x ( m ) N m ) ∈ R N m , 1 ≤ m ≤ M , is expressed b y a determinan t of a P M m =1 N m × P M m =1 N m asymmetric real matrix; ρ sin  t 1 , x (1) N 1 ; . . . ; t M , x ( M ) N M  = det 1 ≤ j ≤ N m , 1 ≤ k ≤ N n 1 ≤ m,n ≤ M h K sin ( t m , x ( m ) j ; t n , x ( n ) k ) i . (1.18) The finite dimensional distributions of the pro ces s (Ξ( t ) , P sin ) ar e determined b y K sin through (1.18). It is exp ected that the correlat io n ke rnel K sin is describ ed b y using the sine function as is the cor r elat io n ke rnel K sin of the stationary distribution µ sin giv en by (1.3 ). It is indeed true. Set f ( z ) = sin( π z ) , z ∈ C , (1.19) and p sin ( t, x ) = e − x 2 / 2 t p 2 π | t | , t ∈ R \ { 0 } , x ∈ C . (1.20) When t > 0 , p sin ( t, y − x ) is the heat k ernel: the solution of the heat equation ∂ u ( t, x ) /∂ t = (1 / 2) ∂ 2 u ( t, x ) /∂ x 2 with lim t → 0 u ( t, x ) dx = δ y ( dx ), and is expre ssed 6 using (1.19) as p sin ( t, y − x ) = 1 2 Z R du e − π 2 u 2 t/ 2 n f ( ux ) f ( uy ) + f ( ux + 1 / 2) f ( u y + 1 / 2) o . F or 0 < s < t , b y setting (1.20), the Chapman-Kolmogoro v e quation Z R dy p sin ( t − s, z − y ) p sin ( s, y − x ) = p sin ( t, z − x ) (1.21) can be extended to Z R dy p sin ( − t, z − y ) p sin ( t − s, y − x ) = p sin ( − s, z − x ) . (1.22) Then K sin ( s, x ; t, y ) is giv en by K f ( s, x ; t, y ) = X a ∈ f − 1 (0) Z √ − 1 R dz √ − 1 p f (0 , a ; s, x ) 1 z − a f ( z ) f ′ ( a ) p f ( t, y ; 0 , z ) − 1 ( s > t ) p f ( t, y ; s, x ) , s, t ≥ 0 , x , y ∈ R (1.23) with setting ( 1 .19 ) and p f ( s, x ; t, y ) = p sin ( t − s, y − x ) with (1.20), where f − 1 (0) denotes the zer o set of t he function f ; f − 1 (0) = { z : f ( z ) = 0 } , f ′ ( a ) = d f ( z ) /dz | z = a , and 1 ( ω ) is the indicator of a condition ω ; 1 ( ω ) = 1 if ω is satisfied, and 1 ( ω ) = 0 otherwise. In this pap er R √ − 1 R dz · means the inte gral on the imaginar y a x is in C from − √ − 1 ∞ to √ − 1 ∞ . The w ell-definedness of the correlation k ernel K sin and th us of the pro cess (Ξ( t ) , P sin ) is guarante ed [13] by the fact that the sine function (1.19) is an entir e function ( i.e. , analytic in the whole complex plane C ), and the or der of gr owth ρ f , whic h is generally defined for an en tire f unction f by ρ f = lim sup r →∞ log log M f ( r ) log r for M f ( r ) = max | z | = r | f ( z ) | , is o ne . (The ty p e define d b y σ f = lim sup r →∞ log M f ( r ) /r ρ f is equal to π for (1.19). That is, the sine function (1.19) is an entir e function of exp onential typ e π [16] ; M f ( r ) ∼ e π r as r → ∞ .) W e c an sho w that K sin ( t, x ; t, y ) K sin ( t, y ; t, x ) dxdy → ξ Z ( dx ) 1 ( x = y ) as t → 0 , since f − 1 (0) = sin − 1 (0) /π = Z . It implies that the initia l configuration (1.17) of the relaxation pro ces s (Ξ( t ) , P sin ) shall b e r ega rded as the p oin t-mass distribution on the ze ro set of the sine function (1.19). Moreo v er, w e sho w ed in [13], b y noting 1 z − a f ( z ) f ′ ( a ) = K sin ( z − a ) , 7 if a ∈ f − 1 (0) = Z and z 6 = a for (1.19), that K sin ( s + θ , x ; t + θ , y ) → K sin ( t − s, y − x ) as θ → ∞ (1.24) with the s o-called extende d sine kernel , K sin ( t, x ) =                    Z 1 0 du e π 2 u 2 t/ 2 cos( π ux ) if t > 0 K sin ( x ) if t = 0 − Z ∞ 1 du e π 2 u 2 t/ 2 cos( π ux ) if t < 0 , (1.25) x ∈ R . The equilibrium dynamics in µ sin , first studied b y Sp ohn [25], has b een sho wn to be determinan t a l with the correlation k ernel (1 .2 5) by Nagao and F orrester [18]. This pro cess is realized in the long-term limit of the relaxation pro cess (Ξ( t ) , P sin ). See also the Dirichle t form approac h b y Osada to the rev ersible pro cess with resp ect to µ sin [20, 21]. No w we set f ( z ) = Ai( z ) , z ∈ C . (1.26) The Airy function Ai( z ), (1.4), is another en tire function, whose order of gro wth is ρ f = 3 / 2 with type σ f = 2 / 3; max | z | = r | f ( z ) | ∼ exp[(2 / 3) r 3 / 2 ] as r → ∞ . F or t > 0, w e consider p Ai ( t, y | x ) = Z R du e ut/ 2 f ( u + x ) f ( u + y ) , x, y ∈ R , (1.27) whic h is the solution of the differen tia l eq uation; ∂ ∂ t u ( t, x ) = 1 2  ∂ 2 ∂ x 2 − x  u ( t, x ) with lim t → 0 u ( t, x ) dx = δ y ( dx ) . The in tegral Z R dz p Ai ( t, z | x ) is given by g ( t, x ) = exp  − tx 2 + t 3 24  . (1.28) W e find, for s < t < 0, g ( s, x ) p Ai ( t − s, y | x ) /g ( t, y ) is equal to the tra nsition proba- bilit y density o f B ( t ) + t 2 4 (1.29) from x a t time s t o y at time t , x, y ∈ R , where B ( t ) , t ∈ [0 , ∞ ) is the one- dimensional standard Bro wnian motion. (See also [6 ] and references therein.) Then 8 for s, t ∈ R , s 6 = t, x, y ∈ C , w e set q ( s, t, y − x ) = p sin  t − s,  y − t 2 4  −  x − s 2 4  = 1 p 2 π | t − s | exp  − ( y − x ) 2 2( t − s ) + ( t + s )( y − x ) 4 − ( t − s )( t + s ) 2 32  , (1.3 0) and as an extension of (1.27) w e define p Ai ( t − s, y | x ) = g ( t, y ) g ( s, x ) q ( s, t, y − x ) = 1 p 2 π | t − s | exp  − ( y − x ) 2 2( t − s ) − ( t − s )( y + x ) 4 + ( t − s ) 3 96  . (1.31) Corresp onding to (1 .2 1 ) and (1 .22), we h a v e the t w o sets of equalities Z R dy q ( s, t, z − y ) q (0 , s, y − x ) = q (0 , t, z − x ) (1.32) Z R dy q ( t, 0 , z − y ) q ( s, t, y − x ) = q ( s, 0 , z − x ) (1.33) and Z R dy p Ai ( t − s, z | y ) p Ai ( s, y | x ) = p Ai ( t, z | x ) (1.34) Z R dy p Ai ( − t, z | y ) p Ai ( t − s, y | x ) = p Ai ( − s, z | x ) (1.35) for 0 < s < t . Let K Ai b e the function giv en by (1.2 3) with setting (1.26) and p f ( s, x ; t, y ) = p Ai ( t − s, y | x ) with (1.31). W e will prov e that K Ai is w ell-defined a s a correlation k ernel and it determines finite dimensional distributions of an infinite particle system through a similar form ula to (1.18). W e denote this system b y (Ξ A ( t ) , P Ai ). The fact that p Ai used in K Ai is a transfor m (1.31) of the transition probability densit y q of (1.29) is the origin of the q uadratic term t 2 / 4 in (1.12). W e can show t ha t K Ai ( t, x ; t, y ) K Ai ( t, y ; t, x ) dxdy → ξ A ( dx ) 1 ( x = y ) as t → 0 . By using t he in tegral form ula for (1.26) 1 z − a f ( z ) f ′ ( a ) = 1 (Ai ′ ( a )) 2 Z ∞ 0 du Ai( u + z )Ai( u + a ) for a ∈ A , z 6 = a , and the fact that { Ai( x + a ) / Ai ′ ( a ) , a ∈ A } forms a complete or- thonormal basis for the space L 2 (0 , ∞ ) o f square integrable functions on the in terv al (0 , ∞ ) [2 7], w e will pro v e that K Ai ( s + θ , x ; t + θ , y ) → K Ai ( t − s, y | x ) as θ → ∞ , (1.36) 9 where K Ai is the so- called extende d A iry kernel , K Ai ( t, y | x ) =            Z ∞ 0 du e − ut/ 2 Ai( u + x )Ai( u + y ) if t ≥ 0 − Z 0 −∞ du e − ut/ 2 Ai( u + x )Ai( u + y ) if t < 0 , (1.37) x, y ∈ R . W e denote b y (Ξ A ( t ) , P Ai ) the infinite particle syste m, whic h is deter- minan ta l with the correlation k ernel K Ai [9, 19, 15, 12]. The Airy k ernel (1.7) of µ Ai is giv en by K Ai ( y | x ) = K Ai (0 , y | x ) and th us (Ξ A ( t ) , P Ai ) is a rev ersible pro- cess with respect to µ Ai . The pro cess (Ξ A ( t ) , P Ai ), whic h is determinantal with the correlation kerne l K Ai , is a non-equilibrium infinite particle system exhibiting the relaxation phenomenon fro m the initial configuration ξ A to t he stationary state µ Ai . Then consider the finite-pa rticle system (1.12) again. Let P ξ N A b e the distri- bution of the pro cess Ξ A ( t ) = P N j =1 δ Y j ( t ) starting f rom a configur a tion ξ N . W e denote b y M the space of nonnegativ e in teger-v alued Radon measu res on R , whic h is a Polish space with the vague top olo g y : w e sa y ξ n con verges to ξ v a guely , if lim n →∞ Z R ϕ ( x ) ξ n ( dx ) = Z R ϕ ( x ) ξ ( dx ) for an y ϕ ∈ C 0 ( R ), where C 0 ( R ) is the set of all con tinuous real-v alued functions with compact supp orts. Any elemen t ξ of M can b e represen ted as ξ ( · ) = P j ∈ Λ δ x j ( · ) with an index set Λ and a sequence of p oin ts in R , x = ( x j ) j ∈ Λ satisfying ξ ( I ) = ♯ { x j : x j ∈ I } < ∞ for any compact subs et I ⊂ R . F o r A ⊂ R , w e write the r estriction of ξ on A as ( ξ ∩ A )( · ) = X j ∈ Λ: x j ∈ A δ x j ( · ). W e put M 0 = n ξ ∈ M : ξ ( { x } ) ≤ 1 for an y x ∈ R o . W e will prov e that the finite particle pro ce ss (Ξ A ( t ) , P ξ N A ) is determinan tal for an y initial configuration ξ N ∈ M 0 and give the correlation k ernel K ξ N A (Prop osition 2.4). F or ξ ∈ M with an infinite n umber of particles ξ ( R ) = ∞ , when K ξ ∩ [ − L,L ] A con verges to a contin uous function a s L → ∞ , the limit is written as K ξ A . If P ξ ∩ [ − L,L ] A con verges to a probabilit y measure P ξ A on M [0 , ∞ ) , whic h is determinan tal with the correlation k ernel K ξ A , we akly in the sense of finite dimensional dis tributions as L → ∞ in the v ague top ology , w e sa y that the pro ces s (Ξ A ( t ) , P ξ A ) is wel l define d wi th the c orr elation kerne l K ξ A . (The regularit y o f the sample paths of Ξ A ( t ) will b e discussed in the forthcoming pap er [14].) W e will giv e sufficien t conditions for initial configurations ξ ∈ M 0 so that the pro ces s (Ξ A ( t ) , P ξ A ) is w ell defined (Theorem 2.5). W e denote by X A the set of configurations ξ satisfying the conditions and put X A 0 = X A ∩ M 0 . It is clear that the configuration ξ A ∈ X A 0 . Then, if w e consider the finite particle sy stems Ξ A ( t ) = P N j =1 δ Y j ( t ) , N ≥ 2, with (1.1 2) starting from the N -particle appro ximation of ξ A , (1.15), w e can prov e (Ξ A ( t ) , P ξ N A A ) → (Ξ A ( t ) , P Ai ) as N → ∞ in the sense of fi- nite dimensional distributions ( Theorem 2.6 (i)). That is, (Ξ A ( t ) , P Ai ) = (Ξ A ( t ) , P ξ A A ) with (1.9). Moreo ver, we will sho w (1 .3 6 ) and pro v e the relaxation phenomenon (Ξ A ( t + θ ) , P Ai ) → (Ξ A ( t ) , P Ai ) (The orem 2.6 (ii)). 10 The pap er is organized as follo ws. In Sect. 2 preliminaries and main results are give n. Some remarks on extensions of the presen t results a re also giv en there. In Sect. 3 the prop erties of the Airy function used in this pap er are summarized. Section 4 is dev oted to pro ofs of results. 2 Preliminaries and Main Result s F or ξ ( · ) = P j ∈ Λ δ x j ( · ) ∈ M , w e in tro duce the follo wing opera t ions ; (shift) for u ∈ R , τ u ξ ( · ) = X j ∈ Λ δ x j + u ( · ), (square) ξ h 2 i ( · ) = X j ∈ Λ δ x 2 j ( · ). W e use the conv en tion suc h that Y x ∈ ξ f ( x ) = exp  Z R ξ ( dx ) log f ( x )  = Y x ∈ supp ξ f ( x ) ξ ( { x } ) for ξ ∈ M a nd a f unction f on R , where supp ξ = { x ∈ R : ξ ( { x } ) > 0 } . F or a m ultiv ariat e symmetric function g w e write g (( x ) x ∈ ξ ) for g (( x j ) j ∈ Λ ). 2.1 Determinan tal p ro cesse s As an M -v alued process ( Ξ( t ) , P ξ ), w e consider the system suc h that, for any inte ger M ≥ 1 , f m ∈ C 0 ( R ) , θ m ∈ R , 1 ≤ m ≤ M , 0 < t 1 < · · · < t M < ∞ , the exp ectation of exp n M X m =1 θ m Z R f m ( x )Ξ( t m , dx ) o can be expanded b y χ m ( x ) = e θ m f m ( x ) − 1 , 1 ≤ m ≤ M , as G ξ [ χ ] ≡ E ξ " exp ( M X m =1 θ m Z R f m ( x )Ξ( t m , dx ) )# = X N 1 ≥ 0 · · · X N M ≥ 0 M Y m =1 1 N m ! Z R N 1 N 1 Y j =1 dx (1) j · · · Z R N M N M Y j =1 dx ( M ) j × M Y m =1 N m Y j =1 χ m  x ( m ) j  ρ  t 1 , x (1) ; . . . ; t M , x ( M )  . (2.1) Here ρ ’s a re lo cally inte grable functions, whic h are symmetric in the sense that ρ ( . . . ; t m , σ ( x ( m ) ); . . . ) = ρ ( . . . ; t m , x ( m ) ; . . . ) with σ ( x ( m ) ) ≡ ( x ( m ) σ (1) , . . . , x ( m ) σ ( N m ) ) for 11 an y p erm utation σ ∈ S N m , 1 ≤ ∀ m ≤ M . In suc h a system ρ ( t 1 , x (1) ; . . . ; t M , x ( M ) ) is called the ( N 1 , . . . , N M )- multitime c orr elation f unc tion and G ξ [ χ ] the gener ating function of multitime c orr elation functions . There are no mu ltiple p oin ts with prob- abilit y one for t > 0. Then w e assume that t here is a function K ( s, x ; t, y ), whic h is con tinuous with respect to ( x, y ) ∈ R 2 for an y fixed ( s, t ) ∈ [0 , ∞ ) 2 , s uc h t ha t ρ  t 1 , x (1) ; . . . ; t M , x ( M )  = det 1 ≤ j ≤ N m , 1 ≤ k ≤ N n 1 ≤ m,n ≤ M " K ( t m , x ( m ) j ; t n , x ( n ) k ) # for a n y in teger M ≥ 1, an y sequence ( N m ) M m =1 of p ositiv e integers, and any time sequence 0 < t 1 < · · · < t M < ∞ . Let T = { t 1 , . . . , t M } . W e note t ha t Ξ T = P t ∈ T δ t ⊗ Ξ( t ) is a determinantal ( F ermion) p oin t pro c ess on T × R with an op erator K giv en b y K f ( s, x ) = X t ∈ T Z R dy K ( s, x ; t, y ) f ( t, y ) , f ( t, · ) ∈ C 0 ( R ) , t ∈ T . When K is symmetric, Soshnik ov [24] and Shirai and T ak ahashi [23] ga v e sufficien t conditions for K to be a correlation k ernel of a determinan t a l point process. Though suc h conditions are not kno wn for asymmetric cases, a v ariet y of pro cesses, whic h are determinan ta l with asymmetric correlation kerne ls, hav e b een studied. See, for example, [29, 12]. If there exists a function K , whic h has the ab ov e prop erties and determines the finite dimens ional distributions of the pro cess (Ξ( t ) , P ξ ), w e sa y the pro cess (Ξ( t ) , P ξ ) is determinantal with the c orr elation kernel K [13]. F or N ∈ N , the determinan t o f an N × N matrix M = ( m j k ) 1 ≤ j,k ≤ N is defined b y X σ ∈S N sgn( σ ) N Y j =1 m j σ ( j ) , where sgn( σ ) denotes t he sign of p erm utation σ . An y p erm utation σ consists of exc lusiv e cycles. If w e write eac h cyclic permutation as c =  a b · · · ω b c · · · a  and the nu m b er o f cyclic p erm utations in a giv en σ as ℓ ( σ ), then the determinan t of M is expressed as det M = X σ ∈S N sgn( σ ) Y c j :1 ≤ j ≤ ℓ ( σ )  m ab m bc . . . m ω a  . It implies that, with giv en a 1 , a 2 , . . . , a N , ev en if eac h elemen t m j k of the mat rix M is replaced by m j k × ( a j /a k ), the v alue of determinan t is not c hanged. The ab o ve observ ation will lead to the follo wing lemma. 12 Lemma 2.1 L et (Ξ( t ) , P ) and ( e Ξ( t ) , e P ) b e the pr o c esses, which ar e d e t erminantal with c orr elation kernels K and e K , r esp e ctively. If ther e is a function G ( s, x ) , which is c ontinuous w ith r esp e ct to x ∈ R fo r any fixe d s ∈ [0 , ∞ ) , such that K ( s, x ; t, y ) = G ( s, x ) G ( t, y ) e K ( s, x ; t, y ) , s, t ∈ [0 , ∞ ) , x, y ∈ R , (2.2) then (Ξ( t ) , P ) = ( e Ξ( t ) , e P ) (2.3) in the sense o f finite d i m ensional distributions. In literatures, (2.2) is called the gauge tr ansforma t ion and (2 .3) is said to be the gauge inv arianc e of the dete rminan t al pro cesses. 2.2 The W eierstrass canonical pro duct and en tire functions F or ξ N ∈ M 0 , ξ N ( R ) = N < ∞ , with p ∈ N 0 ≡ N ∪ { 0 } w e consider the product Π p ( ξ N , z ) = Y x ∈ ξ N ∩{ 0 } c G  z x , p  , z ∈ C , where G ( u, p ) =        1 − u if p = 0 (1 − u ) ex p  u + u 2 2 + · · · + u p p  if p ∈ N . (2.4) The functions G ( u, p ) ar e called the Weiers tr ass primary factors . With α > 0 w e put M α ( ξ N ) =  Z { 0 } c 1 | x | α ξ N ( dx )  1 /α . F or ξ ∈ M 0 with ξ ( R ) = ∞ , w e write M α ( ξ , L ) for M α ( ξ ∩ [ − L, L ]) , L > 0, and put M α ( ξ ) = lim L →∞ M α ( ξ , L ) , if the limit finitely exists. If M p +1 ( ξ ) < ∞ for some p ∈ N 0 , the limit Π p ( ξ , z ) = lim L →∞ Π p ( ξ ∩ [ − L, L ] , z ) = Y x ∈ ξ ∩{ 0 } c G  z x , p  , z ∈ C (2.5) finitely exists. This infinite pro duct is called the Weierstr ass c anonic a l pr o duct of genus p [16]. The Hadamar d the or em [1 6] claims that an y en tire function f of finite order ρ f < ∞ can be represen ted b y f ( z ) = z m e P q ( z ) Π p ( ξ f , z ) , (2.6) 13 where p is a nonnegativ e in teger less than o r equal to ρ f , P q ( z ) is a p olynomial in z o f degree q ≤ ρ f , m is the m ultiplicit y of the ro ot at the origin, and ξ f = P x ∈ f − 1 (0) ∩{ 0 } c δ x . W e giv e t w o examples ; sin( π z ) = π z Π 0 ( ξ Z , z ) , (2.7) Ai( z ) = e d 0 + d 1 z Π 1 ( ξ A , z ) (2.8) with (1.17), (1.9) and d 0 = log Ai(0) = − log  3 2 / 3 Γ(2 / 3)  , d 1 = Ai ′ (0) Ai(0) = − 3 1 / 3 Γ(2 / 3) Γ(1 / 3) = − 3 5 / 6 (Γ(2 / 3)) 2 2 π . (2.9) F or ξ N ∈ M 0 with ξ N ( R ) = N w e put Φ p ( ξ N , a, z ) ≡ Π p ( τ − a ξ N , z − a ) = Y x ∈ ξ N ∩{ a } c G  z − a x − a , p  , a, z ∈ C . (2.10) With (1.15) w e set Φ A ( ξ N , z ) ≡ e d 1 z exp " Z R z x ξ N A ( dx ) # Π 0 ( ξ N , z ) = e d 1 z exp " Z { 0 } c z x ( ξ N A − ξ N )( dx ) # Π 1 ( ξ N , z ) , z ∈ C , (2 .1 1) Φ A ( ξ N , a, z ) ≡ Φ A ( τ − a ξ N , z − a ) = e d 1 ( z − a ) exp " Z R z − a x ξ N A ( dx ) # Φ 0 ( ξ N , a, z ) , a, z ∈ C . (2.12) Lemma 2.2 L et ξ N ∈ M 0 with ξ N ( R ) = N < ∞ and ξ N ( { 0 } ) = 0 . The n fo r a ∈ supp ξ N , z 6 = a , Φ A ( ξ N , a, z ) = 1 z − a Φ A ( ξ N , z ) Φ ′ A ( ξ N , a ) , (2.13) wher e Φ ′ A ( · , a ) = ∂ Φ A ( · , z ) /∂ z    z = a . Pr o of. Since 1 − ( z − a ) / ( x − a ) = ( x − z ) / ( x − a ) = (1 − z /x ) / (1 − a/x ), Φ A ( ξ N , a, z ) = e d 1 z exp  Z R z x ξ N A ( dx )  Π 0 ( ξ N − δ a , z ) e d 1 a exp  Z R a x ξ N A ( dx )  Π 0 ( ξ N − δ a , a ) , (2.14) 14 where the n umerator is equ al to Φ A ( ξ N , z ) / (1 − z /a ). F rom (2.11), w e ha v e ∂ ∂ z Φ A ( ξ N , z ) = d 1 Φ A ( ξ N , z ) + e d 1 z Z R 1 x ξ N A ( dx ) exp  Z R z y ξ N A ( dy )  Π 0 ( ξ N , z ) + e d 1 z exp  Z R z y ξ N A ( dy )  Z R  − 1 x  Π 0 ( ξ N − δ x , z ) ξ N ( dx ) . Since a ∈ supp ξ N is assume d, Φ ′ A ( ξ N , a ) = e d 1 a exp  Z R a x ξ N ( dx )   − 1 a  Π 0 ( ξ N − δ a , a ) . It implies that the denominato r of (2.14 ) is equal to − a Φ ′ A ( ξ N , a ). Then (2.13) is obtained. F or ξ N ∈ M 0 with ξ N ( R ) = N w e put M A ( ξ N ) = Z { 0 } c 1 x ( ξ N A − ξ N )( dx ) . (2.15) F or ξ ∈ M 0 with ξ ( R ) = ∞ w e write M A ( ξ , L ) for M A ( ξ ∩ [ − L, L ]) , L > 0, and put M A ( ξ ) = lim L →∞ M A ( ξ , L ) , if the limit finitely ex ists. F or ξ ∈ M 0 , p ∈ N 0 , a ∈ R , and z ∈ C w e define Φ p ( ξ , a, z ) = lim L →∞ Φ p ( ξ ∩ [ a − L, a + L ] , a, z ) and Φ A ( ξ , a, z ) = lim L →∞ Φ A ( ξ ∩ [ a − L, a + L ] , a, z ) , if the limits finitely exist. W e note that Φ p ( ξ , a, z ) finitely exists and is not iden tically 0, if M p +1 ( τ − a ξ ) < ∞ , and Φ A ( ξ , a, z ) do es and Φ A ( ξ , a, z ) 6≡ 0, if | M A ( τ − a ξ ) | < ∞ and M 2 ( τ − a ξ ) < ∞ . F or ξ ∈ M 0 , a ∈ supp ξ , the followin g equalities will hold, if all the en tries of t he m finitely exist ; Φ 0 ( ξ , a, z ) = Π 0 ( ξ , z )Φ 0 ( ξ ∩ { 0 } c , a, 0)  z a  ξ ( { 0 } ) a a − z , Φ 0 ( ξ ∩ { 0 } c , a, 0) = Π 0 ( ξ ∩ {− a } c , − a )Φ 0 ( ξ h 2 i ∩ { 0 } c , a 2 , 0)2 1 − ξ ( {− a } ) , and then Φ 1 ( ξ , a, z ) = e S ( ξ ,a,z ) Π 1 ( ξ , z )Π 1 ( ξ ∩ {− a } c , − a ) × Φ 0 ( ξ h 2 i ∩ { 0 } c , a 2 , 0)  z a  ξ ( { 0 } ) a a − z , (2.16) where S ( ξ , a, z ) = Z { a } c z − a x − a ξ ( dx ) − Z { 0 } c z x ξ ( dx ) + Z { 0 , − a } c a x ξ ( dx ) . (2.17) 15 Lemma 2.3 F or a ∈ A , z 6 = a 1 z − a Ai( z ) Ai ′ ( a ) = Φ A ( ξ A , a, z ) . (2.18) Pr o of. By (2.8) and the definition (2.11), Ai( z ) = e d 0 Φ A ( ξ A , z ) , z ∈ C . (2.19) As appro ximations of the Airy f unc tion w e introduce functions Ai N ( z ) = e d 0 + d 1 z N Y ℓ =1  1 − z a ℓ  e z /a ℓ , N ∈ N , (2.20) where 0 > a 1 > · · · > a N are the first N zeros of Ai( z ). Since ξ N A = P N j =1 δ a j satisfies the condition of Lemma 2.2, (2.13) with (2.19) and Ai ′ N ( a j ) = e d 0 Φ ′ A ( ξ N A , a j ) giv es 1 z − a j Ai N ( z ) Ai ′ N ( a j ) = Φ A ( ξ N A , a j , z ) , 1 ≤ j ≤ N . T aking N → ∞ , w e ha v e (2.18). 2.3 Statemen t of results F or the solution X ( t ) = ( X 1 ( t ) , X 2 ( t ) , . . . , X N ( t )) of Dyson’s mo del (1 .1 ) with β = 2 with the initia l state X (0) = x , we denote the distribution of the pro cess Ξ( t ) = P N j =1 δ X j ( t ) b y P ξ N with ξ N = P N j =1 δ x j . In [1 3 ] we prov ed that Dyson’s model (1.1 ) with β = 2 starting from an y fix ed configuration ξ N ∈ M is determinan tal with the correlation k ernel K ξ N giv en b y K ξ N ( s, x ; t, y ) = 1 2 π √ − 1 I Γ( ξ N ) dz Z √ − 1 R dw √ − 1 × p sin ( s, x − z ) 1 w − z Y x ′ ∈ ξ N  1 − w − z x ′ − z  p sin ( − t, w − y ) − 1 ( s > t ) p sin ( s − t, y − x ) , (2.21) where Γ( ξ N ) is a closed con t our o n the complex plane C encircling the p oin ts in supp ξ N on R once in the p ositiv e dire ction, and p sin is giv en by (1.20). If ξ N ∈ M 0 , b y p erforming the Cauc h y in t e grals (2.21) is w ritten as K ξ N ( s, x ; t, y ) = Z R ξ N ( dx ′ ) Z √ − 1 R dy ′ √ − 1 p sin ( s, x − x ′ )Φ 0 ( ξ N , x ′ , y ′ ) p sin ( − t, y ′ − y ) − 1 ( s > t ) p sin ( s − t, x − y ) . (2.22) Then the follo wing is obtained for the process (Ξ A ( t ) , P ξ N A ) with Ξ A ( t ) = P N j =1 δ Y j ( t ) , where Y ( t ) = ( Y 1 ( t ) , . . . , Y N ( t )) is giv en b y (1.12). 16 Prop osition 2.4 The p r o c e ss (Ξ A ( t ) , P ξ N A ) , starting fr om any fixe d c onfig u r ation ξ N ∈ M 0 with ξ N ( R ) = N < ∞ , is determinantal with the c orr elation kernel K ξ N A given by K ξ N A ( s, x ; t, y ) = Z R ξ N ( dx ′ ) Z √ − 1 R dy ′ √ − 1 q (0 , s, x − x ′ )Φ A ( ξ N , x ′ , y ′ ) q ( t, 0 , y ′ − y ) − 1 ( s > t ) q ( t, s, x − y ) , (2.23) wher e q is given by (1.30). W e in tro duce the follo wing conditions: ( C.1 ) there ex ists C 0 > 0 suc h that | M A ( ξ ) | < C 0 , ( C.2 ) (i) there exist α ∈ (3 / 2 , 2) and C 1 > 0 suc h that M α ( ξ ) ≤ C 1 , (ii) there ex ist β > 0 and C 2 > 0 su c h t hat M 1 ( τ − a 2 ξ h 2 i ) ≤ C 2 ( | a | ∨ 1) − β for all a ∈ supp ξ . W e denote b y X A the set o f configurations ξ satisfying t he conditions ( C.1 ) and ( C.2 ), and pu t X A 0 = X A ∩ M 0 . Theorem 2.5 If ξ ∈ X A 0 , the pr o c ess (Ξ A ( t ) , P ξ A ) is wel l defin e d with the c orr elation kernel K ξ A ( s, x ; t, y ) = Z R ξ ( dx ′ ) Z √ − 1 R dy ′ √ − 1 q (0 , s, x − x ′ )Φ A ( ξ , x ′ , y ′ ) q ( t, 0 , y ′ − y ) − 1 ( s > t ) q ( t, s, x − y ) . (2.24) In the pro of of this theorem, a useful estimate of Φ A in (2.24) is obtained (Lemma 4.3 (ii)). By virtue of it, w e can see K ξ A ( t, x ; t, y ) K ξ A ( t, y ; t, x ) dxdy → ξ ( dx ) 1 ( x = y ) a s t → 0 (2 .2 5) in the v ague top ology . Then Theorem 2.5 give s an infinite particle system start ing form the configura tion ξ . The main result of the presen t pap er is the following. Theorem 2.6 (i) L et ξ N A ( · ) b e the c on figur ation ( 1.15 ). Then (Ξ A ( t ) , P ξ N A A ) → (Ξ A ( t ) , P Ai ) as N → ∞ in the sense of finite dimensio nal distributions. Her e the pr o c ess (Ξ A ( t ) , P Ai ) is de- terminantal with the c orr elation kernel (1.23) with setting (1. 2 6 ) and p f ( s, x ; t, y ) = p Ai ( t − s, y | x ) with (1.31), that is K Ai ( s, x ; t, y ) = X a ∈ Ai − 1 (0) Z √ − 1 R dz √ − 1 p Ai ( s, x | a ) 1 z − a Ai( z ) Ai ′ ( a ) p Ai ( − t, z | y ) − 1 ( s > t ) p Ai ( s − t, x | y ) . (2.26) 17 (ii) L et (Ξ A ( t ) , P Ai ) b e the pr o c ess, which is determinantal with the extende d Airy kernel (1.37). Then (Ξ A ( t + θ ) , P Ai ) → (Ξ A ( t ) , P Ai ) as θ → ∞ (2.27) we akly in the sens e of finite dimensi o nal distribut ions. 2.4 Remarks on Extensions of the Results (1) By de finition (2.15), M A ( ξ A ) = 0. The asymptotic prop ert y of the zeros (1.11) implies ζ A ( α ) ≡  M α ( ξ A )  α = X a ∈A 1 | a | α < ∞ , if α > b ρ f = 3 2 . (2.28) In general, order o f grow th ρ f of a canonical pro duce (2.5) is equal to t he c onver genc e exp onent b ρ f of the sequence of its zeros [16]. F or Ai( z ), ρ f = 3 / 2. The function ζ A ( α ) ma y b e called the Airy zeta function [30], whic h is meromorphic in the whole of C [7]. Moreo v er, w e know ζ A (2) = M 1 ( ξ h 2 i A ) = X a ∈A 1 a 2 = d 2 1 < ∞ (2.29) with (2.9). Then ξ A satisfies the conditions ( C .1 ) and ( C.2 ) : ξ A ∈ X A . Since ξ A ∈ M 0 , Theorem 2.5 guarante es t he w ell-definedness of the infinite particle system (Ξ A ( t ) , P ξ A A ). (Its equiv a le nce with (Ξ( t ) , P Ai ) is stated in Theorem 2 .6 (i).) Note that the negativ e div ergence (1.16 ) of the drift term D A N t of (1.12) in N → ∞ f o r t < ∞ corresp onds to that ζ A (1) = − P a ∈A (1 /a ) = ∞ . This fa ct and (2.29) mean that the Airy f unction has genus 1 [16 , 30]. Examples o f infinite particle configura t io ns in X A 0 other tha n ξ A are given as follo ws. F or κ > 0, w e put g κ ( x ) = sgn( x ) | x | κ , x ∈ R , and η κ ( · ) = X ℓ ∈ Z δ g κ ( ℓ ) ( · ) . F or any κ > 1 / 2 w e can confirm b y simple calculation that any configuration ξ ∈ M 0 with supp ξ ⊂ supp η κ = { g κ ( ℓ ) : ℓ ∈ Z } satisfie s ( C.2 )(i) with an y α ∈ ( 1 /κ, 2) and some C 1 = C 1 ( α ) > 0 dep ending o n α and ( C.2 )(ii) with an y β ∈ (0 , 2 κ − 1) and some C 2 = C 2 ( β ) > 0 dep ending on β . As sume that ξ ∈ M 0 is c hosen so that supp ξ ⊂ supp η κ for some κ > 1 / 2 and | M A ( ξ ) | < ∞ . Then ξ ∈ X A 0 . The fact (1.11) implies that this assumption can b e satisfied only if κ ∈ (1 / 2 , 2 / 3]. (2) If there exists , ho w eve r, β ′ < ( β − 1) ∧ ( β / 2) for ξ ∈ M 0 suc h t hat ♯ { x ∈ ξ : ξ ([ x − | x | β ′ , x + | x | β ′ ]) ≥ 2 } = ∞ , then ξ do es not satisfy the condition ( C.2 ) (ii). In order to include suc h initial configurations as w ell as those with multiple p oin ts in 18 our study of the pro cess (Ξ A ( t ) , P ξ A ) with ξ ( R ) = ∞ , w e intro duce another condition for configurations: ( C.3 ) there ex ists κ ∈ (1 / 2 , 2 / 3] and m ∈ N suc h that m ( ξ , κ ) ≡ max k ∈ Z ξ  [ g κ ( k ) , g κ ( k + 1)]  ≤ m. W e denote by Y A κ,m the set of configurations ξ satisfying ( C.1 ) and ( C.3 ) with κ ∈ (1 / 2 , 2 / 3] and m ∈ N , and put Y A = [ κ ∈ (1 / 2 , 2 / 3] [ m ∈ N Y A κ,m . Noting that the set { ξ ∈ M : m ( ξ , κ ) ≤ m } is relativ ely compact for each κ ∈ (1 / 2 , 2 / 3] and m ∈ N , w e see that Y A is lo cally compact. In the presen t pap er, w e rep ort our study of the relaxation pro cess (Ξ( t ) , P Ai ) from a sp ecial initial configura tion ξ A to the stat io nary state µ Ai . W e expect tha t µ Ai is an attractor in the configuratio n space Y A and ξ A is a po in t included in the basin. Motiv ated b y suc h consideration, w e are in terested in the con tinuit y of the pro cess with resp ec t to initial configuration. W e ha v e found, how ev er, that if ξ ( R ) = ∞ , the w eak con ve rgence of pro cesses in the sense of finite dimensional distributions can not b e concluded from the con v ergence of initial configura tions in the v ague top ology . F ollow ing the idea give n by o ur previous pap er [13], w e intro duc e a s tronger top ology for Y A . Supp ose that ξ , ξ n ∈ Y A , n ∈ N . W e say that ξ n con verges Φ A - mo der ately to ξ , if lim n →∞ Φ A ( ξ n , √ − 1 , · ) = Φ A ( ξ , √ − 1 , · ) uniformly on an y compact set of C . (2.30) It is e asy to see that (2.30) is satisfied, if the follow ing t wo conditions hold: lim L →∞ sup n> 0 lim M →∞      M A ( ξ n , M ) − M A ( ξ n , L )      = 0 , (2.31) lim L →∞ sup n> 0      M 2 ( ξ n ) − M 2 ( ξ n , L )      = 0 . (2.32) By the similar argument giv en in [13], the following stateme n ts a r e prov ed. (i) If ξ ∈ Y A , the pro cess (Ξ A , P ξ A ) is w ell define d. (ii) Suppose that ξ , ξ n ∈ Y A κ,n , n ∈ N , for some κ ∈ (1 / 2 , 2 / 3] a nd m ∈ N . If ξ n con verges Φ A -mo derately to ξ , then (Ξ A , P ξ n A ) → (Ξ A , P ξ A ) w eakly in the sense of finite dimens ional distributions as n → ∞ in the v ague top ology . Moreo ver, w e can sho w µ Ai ( Y A ) = 1. By this fact and the ab o ve me n t ioned con tinuit y with resp e ct to initial configura tions, we can prov e that the stationary 19 pro cess ( Ξ A ( t ) , P Ai ), whic h is determinan ta l with the extended Airy ke rnel ( 1 .37 ), is Markovian [14]. (3) As men tioned in Intro duc tion, the purp ose of the presen t pap er is to giv e a metho d for asymmetric initial configurations to construct infinite particle systems of Bro wnian motions interacting through pair force 1 /x . In o r der to clarify the results, w e hav e concen trated on the case in this pa p er suc h that the initial configuration is ξ A (Theorem 2.6 ) or its mo dification ξ ∈ X A 0 ; see the condition (C.1) with (2.15) (Theorem 2.5). In the former case the constructed infinite particle system (Ξ A ( t ) , P Ai ) has the stationary measure µ Ai , whic h is obtained in the soft-edge scaling limit o f the eigen v alue distribution in GUE w ell-studied in the r andom matrix theory . Th us w e ha v e sp ecified the entire function u sed in our a nalys is in the form Φ A ( ξ , z ) giv en b y (2.11), whic h is suitable for the Airy function (see (2.19)). The p oin t of our metho d is to put the relationship b et w een the entire function app earing in the correlation k ernel K ξ , the “t ypical” initial configuration ξ f , and the drift term in the SDEs providing finite-particle appro ximations. By the same argumen t as rep orted here, the follo wing will be prov ed. Let f b e the en tire function suc h t ha t f (0 ) 6 = 0, it is expressed by the W eierstrass canonical pro duct of genus one, Π 1 ( ξ f , z ), and the zeros can b e lab elled as 0 < | x 1 | < | x 2 | < · · · . Then with ξ f = P ∞ j =1 δ x j w e put D ( ξ f , N ) = N X j =1 1 x j and in tro duce the N - pa r ticle s ystem Y f j ( t ) = X j ( t ) + D ( ξ f , N ) t, 1 ≤ j ≤ N , t ∈ [0 , ∞ ) , where X ( t ) = ( X 1 ( t ) , . . . , X N ( t )) is the solution of Dyson’s mo del (1.1) starting from the first N zeros of f , X j (0) = x j , 1 ≤ j ≤ N . Then Ξ f ( t ) = P N j =1 δ Y f j ( t ) con verges to the dynamics in N → ∞ , in the sens e of finite dim ensional distribution, whic h is determinan tal with the correlation k ernel K ξ f f ( s, x ; t, y ) = Z R ξ f ( dx ′ ) Z √ − 1 R dy ′ √ − 1 p sin ( s, x − x ′ )Φ 1 ( ξ f , x ′ , y ′ ) p sin ( − t, y ′ − y ) − 1 ( s > t ) p sin ( s − t, x − y ) . (2.33) Moreo ver, ev en if t he initial configuration ξ is differen t fr om ξ f , but it satisfies the condition     Z L − L 1 x ( ξ f − ξ )( dx )     < C 0 for any L > 0 with a p ositiv e finite C 0 indep en den t o f L , then the pro cess starting from ξ , is w ell- defined. In general, the obtained dynamics with an infinite n umber of particles is not stationary , while Theorem 2.6 gav e the example whic h con verges to a s tationary dynamics (Ξ A ( t ) , P Ai ) in the lo ng -term limit. 20 3 Prop ertie s of th e Airy F unc tions 3.1 In tegrals By the f act Ai ′′ ( x ) = x Ai( x ), the follo wing primitiv e is obtained for c 6 = 0 [30], Z du (Ai( c ( u + x ))) 2 = ( u + x )(Ai( c ( u + x ))) 2 − 1 c (Ai ′ ( c ( u + x ))) 2 , (3.1 ) Z du Ai( c ( u + x ))Ai( c ( u + y )) = Ai ′ ( c ( u + x ))Ai( c ( u + y )) − Ai( c ( u + x ))Ai ′ ( c ( u + y )) c 2 ( x − y ) . (3.2) By setting c = 1 and in tegral inte rv al b e [0 , ∞ ) in (3.2) , w e o bta in the in tegral Z ∞ 0 du Ai( u + x )Ai( u + y ) = Ai( x )Ai ′ ( y ) − Ai ′ ( x )Ai( y ) x − y , since lim x →∞ Ai( x ) = lim x →∞ Ai ′ ( x ) = 0 b y (1.5). If w e set y = a ∈ A a nd x = z 6 = a , then Z ∞ 0 du Ai( u + z )Ai( u + a ) = Ai( z )Ai ′ ( a ) z − a , since Ai( a ) = 0. Then w e ha v e the expre ssion 1 z − a Ai( z ) Ai ′ ( a ) = 1 (Ai ′ ( a )) 2 Z ∞ 0 du Ai( u + z )Ai( u + a ) (3.3) for a ∈ A , z 6 = a . 3.2 Airy transform The follo wing in tegral form ulas ar e pro v ed [11]. Lemma 3.1 F or c > 0 , x, y ∈ R Z R du e cu Ai( u + x )Ai ( u + y ) = 1 √ 4 π c e − ( x − y ) 2 / (4 c ) − c ( x + y ) / 2+ c 3 / 12 , (3.4) Z R dy Z R du e cu Ai( u + x )Ai( u + y ) = e − cx + c 3 / 3 . (3.5) Pr o of. Consider the in tegral I = Z R du e cx Ai( u + x )Ai( u + y ) = Z ∞ 0 du e cu Ai( u + x )Ai( u + y ) + Z 0 −∞ du e cu Ai( u + x )Ai ( u + y ) . 21 By the definition o f the Airy function (1.4), fo r an y η > 0, I = Z ∞ 0 du e cu 1 (2 π ) 2 Z ℑ z = η dz Z ℑ w >c − η dw e √ − 1 { z 3 / 3+( u + x ) z + w 3 / 3+( y + u ) w } + Z 0 −∞ du e cu 1 (2 π ) 2 Z ℑ z = η dz Z ℑ w c − η dw e √ − 1( z 3 / 3+ xz + w 3 / 3+ yw ) Z ∞ 0 du e { c + √ − 1( z + w ) } u + 1 (2 π ) 2 Z ℑ z = η dz Z ℑ w c − η dw e √ − 1( z 3 / 3+ xz + w 3 / 3+ yw ) 1 c + √ − 1( z + w ) + 1 (2 π ) 2 Z ℑ z = η dz Z ℑ w c − η dw  e √ − 1( w 3 / 3+ yw ) c + √ − 1( z + w ) . It is e qual to the integral 1 2 π Z ℑ z = η dz e √ − 1( z 3 / 3+ xz ) 1 2 π √ − 1 I C dw e √ − 1( w 3 / 3+ √ − 1 yw ) w − ( √ − 1 c − z ) , where C is a closed contour on C encircling a p ole at w = √ − 1 c − z o nc e in the p ositiv e direction. By p erforming the C auc hy in tegral, we hav e I = 1 2 π Z ℑ z = η dz e √ − 1( z 3 / 3+ xz ) e √ − 1( √ − 1 c − z ) 3 / 3 − y ( c + √ − 1 z ) = 1 2 π e c 3 / 3 − cy Z ℑ z = η dz e − cz 2 + √ − 1( x − y + c 2 ) z . By perfor min g a Gaussian integral, w e ha v e (3.4). Since − 1 4 c ( x − y ) 2 − c 2 ( x + y ) + c 3 12 = − 1 4 c n y − ( x − c 2 ) o 2 − cx + c 3 3 , the Gaussian in tegration of (3.4) w ith respect to y give s (3.5). By setting c = t/ 2 > 0 in (3.5) and c = ( t − s ) / 2 > 0 in (3.4), resp ectiv ely , w e obtain the equalities Z R dy Z R du e ut Ai( u + x )Ai( u + y ) = g ( t, x ) , Z R du e u ( t − s ) / 2 Ai( u + x )Ai ( u + y ) = g ( t, y ) g ( s, x ) q ( s, t, y − x ) 22 with (1.28) and (1.30). Th us w e hav e defined g ( t, x ) for any t ∈ R b y (1 .2 8) and p Ai ( s, x ; t, y ) for an y s, t ∈ R , s 6 = t b y (1.31). As special cases, w e ha v e p Ai ( t, y | x ) = g ( t, y ) q (0 , t, y − x ) , (3.6) p Ai ( − t, y | x ) = 1 g ( t, x ) q ( t, 0 , y − x ) , t > 0 , x, y , ∈ R . (3.7) If w e tak e the c → 0 limit in (3.5), w e obtain Z R dξ Z R dx Ai( ξ − x )Ai ( ξ ′ − x ) = 1 . The expression (3.4) and the ab o v e r esult implies the orthonormality of the Airy function in the sense;  Z R du Ai( u + x )Ai( u + y )  dy = δ x ( dy ) . (3.8) The A i ry tr ansform f ( x ) 7→ ϕ ( ξ ) is t hen defined b y ϕ ( ξ ) = Z R dx f ( x )Ai( ξ + x ) , (3.9) and the in ve rse transform is given b y f ( x ) = Z R dξ ϕ ( ξ )Ai( ξ + x ) . No w a parameter c ∈ C is in tro duced a nd t he family of functions are defined as { w c ( x ) = Ai( x/c ) / | c |} . The Airy transform (3.9) is then generalized as ϕ c ( ξ ) = Z R dξ f ( x ) w c ( ξ + x ) = 1 | c | Z R dxf ( x )Ai  ξ + x c  . Lemma 3.2 The Airy tr ansfo rm with c of the normalize d Gaussian function f ( x ) = e − x 2 / √ π is giv e n by ϕ c ( ξ ) = | c | − 1 e { ξ +1 / ( 24 c 3 ) } / (4 c 3 ) Ai( ξ /c + 1 / (16 c 4 )) . T hat is, Z R dx 1 √ π e − x 2 Ai  ξ + x c  = exp  1 4 c 3  ξ + 1 24 c 3  Ai  ξ c + 1 16 c 4  . (3.10) Pr o of. By the definition of the Airy function (1.4), I = Z R dx 1 √ π e − x 2 Ai  ξ + x c  = Z R dx 1 √ π e − x 2 1 2 π Z R dk e √ − 1 { k 3 / 3+( ξ + x ) k/c } = 1 2 π Z R dk e √ − 1 k 3 / 3+ √ − 1 ξ k/c 1 √ π Z R dx exp  − x 2 − √ − 1 k c x  . 23 By perfor min g the Gaussian in tegral w e ha v e I = 1 2 π Z R dk exp  √ − 1 k 3 3 + √ − 1 ξ k c − k 2 4 c 2  . By completing a cub e, w e find the eq ualit y √ − 1 k 3 3 + √ − 1 ξ k c − k 2 4 c 2 = √ − 1 1 3  k + √ − 1 1 4 c 2  3 + √ − 1  ξ c + 1 16 c 4   k + √ − 1 1 4 c 2  + 1 4 c 3  ξ + 1 24 c 3  . By using t he definition of the Airy function (1.4), (3.10) is obtained. F or t > 0 , y , u ∈ R w e will obtain the e qualit y Z √ − 1 R dz √ − 1 Ai( u + z ) q ( t, 0 , z − y ) = Z R dx 1 √ π e − x 2 Ai  √ − 2 t x + u + y − t 2 4  b y c hanging the integral v ariable a s z 7→ x = ( z − y + t 2 / 4) / √ − 2 t . If we set ξ = ( u + y − t 2 / 4) / √ − 2 t and c = 1 / √ − 2 t , the RHS is iden t ified with the LHS of (3.10). Since ξ c + 1 16 c 4 = u + y , 1 4 c 3  ξ + 1 24 c 3  =  − ty 2 + t 3 24  − ut 2 , (3.10) of Lem ma 3.2 with (1.28) give s Z √ − 1 R dz √ − 1 Ai( u + z ) q ( t, 0 , z − y ) = g ( t, y ) e − ut/ 2 Ai( u + y ) , t > 0 , y , u ∈ R . Com bination with (3.7) giv es Z √ − 1 R dz √ − 1 Ai( u + z ) p Ai ( − t, z | y ) = e − ut/ 2 Ai( u + y ) , t > 0 , y , u ∈ R (3.11) 3.3 F ourier-Airy series Let us c onsider the in tegral I ℓℓ ′ = Z ∞ 0 dx Ai( x + a ℓ )Ai( x + a ℓ ′ ) , a ℓ , a ℓ ′ ∈ A . In the c ase ℓ 6 = ℓ ′ , the for mula (3.2) gives I ℓℓ ′ = Ai ′ ( a ℓ )Ai( a ℓ ′ ) − Ai ( a ℓ )Ai ′ ( a ℓ ′ ) a ℓ − a ℓ ′ = 0 , 24 whereas if ℓ = ℓ ′ , the form ula (3.1) giv es I ℓℓ = (Ai ′ ( a ℓ )) 2 . Therefore the f unc tions  Ai( x + a ℓ ) Ai ′ ( a ℓ ) , ℓ ∈ N  (3.12) form an orthogonal basis fo r f ∈ L 2 (0 , ∞ ) (see Sect. 4.12 in [27]). The completeness of (3.12) is also established : X ℓ ∈ N Ai( x + a ℓ )Ai( y + a ℓ ) (Ai ′ ( a ℓ )) 2 dy = δ x ( dy ) , x, y ∈ (0 , ∞ ) . (3.1 3) Then for an y f ∈ L 2 (0 , ∞ ), we can write the express ion f ( x ) = X ℓ ∈ N c ℓ Ai( x + a ℓ ) Ai ′ ( a ℓ ) , x ∈ [0 , ∞ ) , and call it the F ourier-A iry series exp ansion . The co efficien ts c ℓ of this expansion are determined b y c ℓ = { R ∞ 0 dx f ( x )Ai( x + a ℓ ) } / Ai ′ ( a ℓ ). 4 Pro of of Results 4.1 Pro of of Prop osition 2.4 With (1.13) w e put b g ( s, x ) = exp  − D A N  D A N s 2 + s 2 4 − x  , s, x ∈ R . By the definition ( 1.30) of q , for s > 0 , x, x ′ ∈ R , w e hav e p sin ( s, ( x − D A N s − s 2 / 4) − x ′ ) q (0 , s, x − x ′ ) = exp " − 1 2 s (  x − D A N s − s 2 4 − x ′  2 −  x − s 2 4 − x ′  2 )# = exp  − D A N s  D A N s 2 + s 2 4 − x + x ′  = b g ( s, x ) e − D A N x ′ , and for t > 0 , y , y ′ ∈ R , w e ha v e p sin ( − t, y ′ − ( y − D A N t − t 2 / 4)) q ( t, 0 , y ′ − y ) = 1 b g ( t, y ) e D A N y ′ . 25 Similarly , w e hav e p sin ( s − t, ( x − D A N s − s 2 / 4) − ( y − D A N t − t 2 / 4)) q ( t, s, y − x ) = exp " − 1 2( s − t ) (  x − y − D A N ( s − t ) − s 2 − t 2 4  2 −  x − s 2 4  −  y − t 2 4  2 )# = exp  − D A N  D cA N 2 ( s − t ) + s 2 − t 2 4 − ( x − y )  = b g ( s, x ) b g ( t, y ) . Then w e ha ve K ξ N  s, x − D A N s − s 2 4 ; t, y − D A N t − t 2 4  = b g ( s, x ) b g ( t, y )  Z R ξ N ( dx ′ ) Z √ − 1 R dy ′ √ − 1 × q (0 , s, x − x ′ ) e − D A N x ′ Φ 0 ( ξ N , x ′ , y ′ ) e D A N y ′ q ( t, 0 , y ′ − y ) − 1 ( s > t ) q ( t, s, x − y ) # . The iden tity (2.12) implies e − D A N x ′ Φ 0 ( ξ N , x ′ , y ′ ) e D A N y ′ = Φ A ( ξ N , x ′ , y ′ ) . By t he gauge in v ariance of de terminan ta l pro cesses (Lemma 2.1), the proof is com- pleted. 4.2 Pro of of Theorem 2.5 First w e prepare some lemm as for pro ving Theorem 2.5. Lemma 4.1 L et α ∈ (1 , 2) and δ > α − 1 . S u pp ose that M α ( ξ ) < ∞ and put L 0 = L 0 ( α, δ , ξ ) = (2 M α ( ξ )) α/ ( δ − α +1) . Th en M 1 ( ξ , L ) ≤ L δ , L ≥ L 0 . Since this lem ma w as prov ed as Lemma 4.3 in [13], here w e omit the proof. Lemma 4.2 If ξ satisfie s ( C.2 ) (i) and (ii) , for any θ ∈ ( α ∨ (2 − β ) , 2) ther e exists C = C ( C 1 , C 2 , θ ) > 0 such that     Z { 0 ,a } c  1 x − 1 x − a  ξ ( dx )     ≤ C | a ∨ 1 | θ − 1 , a ∈ supp ( ξ − δ 0 ) . (4.1) 26 Pr o of. W e divide { 0 , a } c in to three sets A 1 = { x ∈ { 0 , a } c : | x | ≤ | a | / 2 } , A 2 = { x ∈ { 0 , a } c : | a | / 2 < | x | ≤ 2 | a |} , and A 3 = { x ∈ { 0 , a } c : 2 | a | < | x |} , and put I j = Z A j     1 x − 1 x − a     ξ ( dx ) j = 1 , 2 , 3 . When x ∈ A 1 , | x 2 − a 2 | ≥ 3 a 2 / 4 and | x + a | ≤ 3 | a | / 2, and then I 1 = Z A 1 | a || x + a | | x || x 2 − a 2 | ξ ( dx ) ≤ 2 M 1  ξ , | a | 2  . By Lemma 4 .1 f or any δ > α − 1 , we can tak e C > 0 suc h that I 1 ≤ C | a ∨ 1 | δ . (4.2) When x ∈ A 2 , | x + a | ≤ 3 | a | and | a | / | x | ≤ 2, and then I 2 = Z A 2 | a || x + a | | x || x 2 − a 2 | ξ ( dx ) ≤ 6 | a | M 1 ( τ − a 2 ξ h 2 i ) . F rom the c ondition ( C.2 ) (ii) I 2 ≤ 6 C 2 | a ∨ 1 | 1 − β . (4.3) When x ∈ A 3 , | x − a | > | x | / 2, and then I 3 = Z A 3 | a | | x || x − a | ξ ( dx ) ≤ 2 α − 1 | a | α − 1 M α ( ξ ) α . F rom the c ondition ( C.2 ) (i) I 3 ≤ 2 α − 1 C 1 | a ∨ 1 | α − 1 . (4.4) Com bining the estimates (4.2), (4.3) and (4 .4), w e ha v e (4.1). Lemma 4.3 (i) If ξ satisfies the c ond it ions ( C.2 ) ( i) and (ii) , for any θ ∈ ( α ∨ (2 − β ) , 2) t her e e x i s t s C = C ( C 1 , C 2 , θ ) > 0 such that | Φ 1 ( ξ , a, √ − 1 y ) | ≤ exp  C { ( | y | θ ∨ 1) + ( | a | θ ∨ 1) }  , (4.5) for y ∈ R and a ∈ supp ξ . (ii) If ξ satisfies the c onditions ( C.1 ), ( C.2 ) (i) and (ii) , for a n y θ ∈ ( α ∨ (2 − β ) , 2) ther e exists C = C ( C 0 , C 1 , C 2 , θ ) > 0 such that | Φ A ( ξ , a, √ − 1 y ) | ≤ exp  C { ( | y | θ ∨ 1) + ( | a | θ ∨ 1) }  , (4.6) for y ∈ R and a ∈ supp ξ . 27 Pr o of. W e pro v e (i) of this lemma. F rom t he condition ( C.1 ) and the r e lation (2.12), (ii) is easily deriv ed fro m (i). W e first consider the case that a = 0 ∈ supp ξ . Remind that Φ 1 ( ξ , 0 , z ) = Π 1 ( ξ , z ) = exp  Z { 0 } c n log  1 − z x  + z x o ξ ( dx )  . When 2 | z | < | x | , b y using the expansion log  1 − z x  = − X k ∈ N 1 k  z x  k , w e hav e    log  1 − z x  + z x    ≤    z x    2 . Then | Π 1 ( ξ ∩ [ − 2 | z | , 2 | z | ] c , z ) | ≤ exp  | z | 2 Z | x | > 2 | z | 1 x 2 ξ ( dx )  ≤ exp n | z | α M α ( ξ ) α o . (4.7) On the other hand | 1 − z /x | ≤ e | z | / | x | . Then | Π 1 ( ξ ∩ [ − 2 | z | , 2 | z | ] , z ) | ≤ exp  2 | z | Z | x |≤ 2 | z | 1 | x | ξ ( dx )  = exp n 2 | z | M 1 ( ξ , 2 | z | ) o . (4.8) F rom (4.7) and (4 .8), with the condition ( C.2 ) (i) and Lemma 4.1, w e see that fo r an y θ ∈ ( α ∨ (2 − β ) , 2) there e xists C = C ( C 1 , θ ) > 0 suc h that | Φ 1 ( ξ , 0 , z ) | ≤ exp  C { ( | z | θ ∨ 1) }  , (4.9) for z ∈ C and a ∈ supp ξ . Next we consider the case that a ∈ supp ξ and a 6 = 0. By the conditions ( C .1 ) and ( C.2 ) the equalit y (2.16) is v alid. By (4.9) | Π 1 ( ξ , z )Π 1 ( ξ ∩ {− a } c , − a ) | ≤ exp h C { ( | z | θ ∨ 1) + ( | a | θ ∨ 1) } i . By the conditio n ( C.2 ) (ii) | Φ 0 ( ξ h 2 i ∩ { 0 } c , a 2 , 0) | ≤ exp n | a | 2 M 1 ( τ − a 2 ξ h 2 i ) o ≤ exp n C 2 ( | a | ∨ 1) 2 − β o , and | ( √ − 1 y /a ) ξ ( { 0 } ) a/ ( a − √ − 1 y ) | ≤ 1. Now we ev aluate S ( ξ , a, z ) . S ( ξ , a, z ) = Z { 0 ,a } c  z − a x − a − z − a x  ξ ( dx ) + ( z − a ) − a ξ ( { 0 } ) − z a + a a − a − a ξ ( {− a } ) = ( z − a ) Z { 0 ,a } c  1 x − a − 1 x  ξ ( dx ) − (1 + ξ ( { 0 } )) z a + 1 + ξ ( { 0 } ) + ξ ( {− a } ) . 28 F rom Lemma 4.2 and the fact 1 /a 2 ≤ C 2 and then | 2 z /a | ≤ 2 √ C 2 | z | , w e ha v e | S ( ξ , a, z ) | ≤ C | z − a || a ∨ 1 | θ − 1 + 2 p C 2 | z | + 3 ≤ C ′ n ( | y | θ ∨ 1) + ( | a | θ ∨ 1) o for some C ′ > 0. This completes the pro of. Pr o of of T he or em 2.5. Note tha t ξ ∩ [ − L, L ], L > 0 and ξ satisfy ( C .1 ) and ( C .2 ) with the same constan ts C 0 , C 1 , C 2 and indices α , β . By virtue of Lemma 4.3 (ii) w e see that the re exists C > 0 suc h that | Φ A ( ξ ∩ [ − L, L ] , a, √ − 1 y ) | ≤ exp h C n ( | y | ∨ 1) θ + ( | a | ∨ 1) θ oi , ∀ L > 0 , ∀ a ∈ supp ξ , ∀ y ∈ R . Since for a n y y ∈ R Φ A ( ξ ∩ [ − L, L ] , a, √ − 1 y ) → Φ A ( ξ , a, √ − 1 y ) , L → ∞ , w e can apply L eb esgue’s c onver genc e the or em to (2.23) and o btain lim L →∞ K ξ ∩ [ − L,L ] A ( s, x ; t, y ) = K ξ A ( s, x ; t, y ) . Since for an y ( s, t ) ∈ (0 , ∞ ) 2 and an y finite in terv al I ⊂ R sup x,y ∈ I    K ξ ∩ [ − L,L ] A ( s, x ; t, y )    < ∞ , w e can obtain the conv ergence of gene rating functions for multitime correlat io n functions (2.1); G ξ ∩ [ − L,L ] [ χ ] → G ξ [ χ ] as L → ∞ . It implies P ξ ∩ [ − L,L ] A → P ξ A as L → ∞ in the s ense of finite dim ensional distributions. Then the pro of is completed. 4.3 Pro of of Theorem 2.6 (i) It is clear that ξ A is an elemen t of X 0 A . (See the item (1) of Sect. 2.4.) Then b y Theorem 2.5 (Ξ A ( t ) , P ξ N A A ) → (Ξ A ( t ) , P ξ A A ) as N → ∞ in t he sense of finite di- mensional distributions, where (Ξ A ( t ) , P ξ A A ) is the determinan tal with the cor r elat io n k ernel K ξ A A ( s, x ; t, y ) = Z R ξ A ( da ) Z √ − 1 R dz √ − 1 q (0 , s, x − a )Φ A ( ξ A , a, z ) q ( t, 0 , z − y ) − 1 ( s > t ) q ( t, s, x − y ) . Using the e qualities (3.6) and (3.7) and the definition (1.31), w e ha ve K ξ A A ( s, x ; t, y ) = Z R ξ A ( da ) Z √ − 1 R dz √ − 1 p Ai ( s, x | a ) g ( s, x ) 1 z − a Ai( z ) Ai ′ ( a ) g ( t, y ) p Ai ( − t, z | y ) − 1 ( s > t ) g ( t, y ) g ( s, x ) p Ai ( s − t, x | y ) = g ( t, y ) g ( s, x ) K Ai ( s, x ; t, y ) 29 with (2.26), where w e hav e used (2.18) of Lemma 2.3. By the ga uge inv ariance, Lemma 2 .1, (Ξ A ( t ) , P ξ A A ) = (Ξ A ( t ) , P Ai ) in the sense of finite dimensional distribu- tions. (ii) If w e use t he expression (3.3), (2.26) becomes K Ai ( s, x ; t, y ) = X a ∈ Ai − 1 (0) Z √ − 1 R dz √ − 1 p Ai ( s, x | a ) × 1 (Ai ′ ( a )) 2 Z ∞ 0 du Ai( u + z )Ai( u + a ) p Ai ( − t, z | y ) − 1 ( s > t ) p Ai ( s − t, x | y ) . By (3.11), the first term of the RHS equals X a ∈ Ai − 1 (0) p Ai ( s, x | a ) 1 (Ai ′ ( a )) 2 Z ∞ 0 du e − ut/ 2 Ai( u + y )Ai( u + a ) . Since s > 0, we can use the expression ( 1 .27 ) f or p Ai ( s, x | a ) and the ab o ve is written as Z ∞ 0 du Z R dw e − ut/ 2+ ws / 2 Ai( u + y )Ai( w + x ) X ℓ ∈ N Ai( u + a ℓ )Ai( w + a ℓ ) (Ai ′ ( a ℓ )) 2 . F rom the c ompleteness (3.13), the abov e giv es K Ai ( s, x ; t, y ) = K Ai ( t − s, y | x ) + R ( s, x ; t, y ) with the e xtended Airy k ernel K Ai giv en by (1.3 7) and R ( s, x ; t, y ) = Z ∞ 0 du Z 0 −∞ dw e − ut/ 2+ ws / 2 Ai( u + y )Ai( w + x ) × X ℓ ∈ N Ai( u + a ℓ )Ai( w + a ℓ ) (Ai ′ ( a ℓ )) 2 . Since for an y fixed s, t > 0 lim θ → ∞ | R ( s + θ, x ; t + θ, y ) | → 0 unifor mly on an y compact subset of R 2 , (1.36) ho lds in the same se nse. Hence we obtain (2 .2 7). This completes the proof. A cknow le dgments. M.K. is sup ported in part b y the Grant -in-Aid for Scie n tific Rese arc h (C) (No.2154 0397) of J a pan So ciet y for the Pr omotion of Science. H.T. is supp orted in part b y the Gran t-in-Aid for Scien tific Researc h (KIBAN-C, No.19540114 ) of Japan So ciet y for the Pr omotion o f S cie nce. 30 References [1] Abramo witz, M., Stegun, I. : Hand bo ok of Mathematical F u nctio ns. Do v er, New Y ork (1965) [2] Adler, M., v an Moer b eke, P . : T he sp ectrum of coupled rand om matrices. Ann. Math. 149 , 921-976 (1999) [3] Adler, M., v an Mo erb ek e, P . : PDF’s for the join t distribu tio ns of the Dyson, Airy and Sine processes. Ann . Probab. 33 , 1 326-13 61 (2005) [4] Dyson, F. J . : A Brownian-motio n mod e l for t he eigenv alues of a random matrix. J. Math. Phys. 3 , 1191-119 8 (1962 ) [5] Dyson, F. J. : The three f old wa y . Algebraic structure of symmetry groups and ensem bles in quan tum mechanics. J. Math. Ph ys. 3 , 1 199-12 15 (196 2) [6] F errari, P . L., Sp ohn, H. : C o nstrained Bro w nian motion : Fluctuations a w a y fr o m circular and parabolic barriers. Ann. Pr obab. 33 , 1302-1325 (2 005) [7] Fla jolet, P ., Louc h ard , G. : An a lytic v ariations on the Airy d istribution. Algorithmica 31 , 361-377 ( 2001) [8] F orrester, P . J. : T h e sp ectrum edge of random matrix ensem b les. Nu cl . Ph ys. B 402 , 709-7 28 (1993 ) [9] F orrester, P . J., Nagao, T., Honner, G. : Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and s o ft edges. Nucl. Phys. B 553 (PM), 601-6 43 (1999 ) [10] Grabiner, D. J.: Bro wnian motion in a W eyl c hamb e r, non-colliding particles, and random matrices. Ann. Inst. Henr i Poincar ´ e, Probab. Stat. 3 5 , 177-204 (1999) [11] Johanss on , K. : Di screte p olyn uclear gro w th and determinant al pro cesses. Comm un. Math. Phys. 242 , 277-329 (2003) [12] Katori, M., T anem u ra, H.: Noncolliding Bro wnian m otion and d etermin antal pro- cesses. J . Stat . Phys. 129 , 123 3-1277 (2007) [13] Katori, M., T anem ur a, H.: Non-equilibr ium dynamics of Dyson’s mo del with an infinite n um b er of particles. Commun. Math. Ph ys. doi:10.1 007/s00 220-0 09-0912-3. arXiv:0812 .4108 [math.PR] [14] Katori, M., T anem ura, H.: in pr eparat ion [15] Katori, M., Nagao, T ., T anem u ra, H. : Infin it e sy s te ms of n o n-colliding Bro wn ia n particles. In: Sto c hastic Analysis on Large S c ale In teracting Systems. Adv. S tud. in Pure Math., v ol. 39, pp.283- 306. Mathematica l Societ y of Japan, T oky o (2004). arXiv:math.PR/030 1143 [16] Levin, B. Y a.: Lectures on Entire F u nctio ns. T ranslations of Mathematica l Mono- graphs, vo l.150. Amer. Math. S oc., Pr o vidence (1996) 31 [17] Meht a, M. L. : Rand o m Matrices, 3rd edn. E lsevier, Amsterdam (2004) [18] Nagao, T., F orrester, P . J. : Multile v el dynamical correlation functions for Dyson’s Bro wn ia n mo tion mo del of r a ndom matrices. P h ys. Lett. A247 , 42-46 (19 98) [19] Nagao, T., Katori, M., H. T anem ur a , H. : Dynamical correlations among vicious random walk ers. Ph ys. Lett. A 307 , 29-35 (2003) [20] Osada, H. : Dirichlet form app roac h to infinite-dimensional Wiener pro cesses w it h singular inte ractions. Comm un . M ath. Phys. 176 , 1 17-131 (1996) [21] Osada, H. : In teracting Bro wnian motions in infinite dimensions with logarithmic in teractio n p oten tials. a rXiv:0902.3 561 [math.PR] [22] Pr¨ ahofer, M., Sp ohn, H. : Scale inv ariance of th e PNG droplet and the Airy p rocess. J. S t at. Phys. 108 , 1071-1106 (200 2) [23] Sh irai, T., T ak ahashi, Y.: Rand om p oin t fields asso ciated with certain F redholm determinan ts I: fermion, P oisson and boson point p rocess. J. F un ct. Anal. 205 , 414- 463 (2003) [24] Soshn ik o v, A. : Determinan tal random p oin t fields. Russian Ma th. Surv eys 5 5 , 9 23- 975 (2000) [25] Sp ohn, H. : Interacting Brownian particles: A study of Dyson’s mo del. In: P a- panicolaou, G. (ed.) Hydro dynamic Beha v ior a nd Inte racting Pa rticle Systems, IMA V olumes in Ma thematics and its App lic ations, v ol. 9, p p .1 51-179 . Springer, Be rlin (1987 ) [26] Sp ohn, H. : Large Scale Dynamics of Interact ing P articles. S p ringer, Berlin (19 91) [27] Titc hmarsh, E. C. : Eigenfun ct ion Expansions Asso ciated w it h Seco nd-order Diffe r- en tial Equations. P art I, 2nd edn. Cla rendon Pr ess, Oxford (1962) [28] T r a cy , C. A., Widom, H. : Lev el-spacing d istributions and the Airy k ernel. Commun. Math. Phys. 159 , 151-174 (1994) [29] T r a cy , C. A., Widom, H. : A system of differen tial equations for the Airy pro cess. Elect. Commun. Probab. 8 , 9 3-98 (2003) [30] V all ´ ee, O., Soares, M. : Airy F unctions and Applications to Physics. Im p erial College Press, London ( 2004) 32