Markov property of determinantal processes with extended sine, Airy, and Bessel kernels

Mark o v prop ert y of determinan tal pro cesses with extended sine, Airy , and Bessel k ernels Mak oto Kator i ∗ and Hidek i T anemura † 22 June 2 011 Abstract When the num b er of particles is finite, the noncolliding Bro w nian motion (the Dyson mo del) and the noncolliding squared Bessel pro cess are determinant al diffusion pro cesses for an y deterministic initial configuration ξ = P j ∈ Λ δ x j , in the sense that an y m u ltitime correlation function is giv en b y a determinant associated with the correlation k ern el, whic h is sp ecified b y an en tire function Φ ha ving zeros in supp ξ . Using such en tire functions Φ, we define new topologies calle d the Φ-mo d erate top ologies. Then we construct three infinite-dimensional determinan tal pr o cesses, as the limits of sequences of determinan tal d iffu sion p ro cesses with fin ite num b ers of p articles in the sense of finite dimensional distributions in the Φ-mo derate top ologies, so that the probabilit y distributions are con tinuous w ith resp ect to initial configur ations ξ with ξ ( R ) = ∞ . W e sho w that our three infinite p article systems are v ersions of th e determinan tal pro cesses with the extended sine, Bessel, and Airy k ern els, resp ectiv ely , which are reve rsible with resp ect to the determinantal p oin t pro cesses obtained in the bu lk scaling limit and the soft-edge scaling limit of the eige n v alue distributions of the Gaussian unitary ensem ble, and the hard-edge scaling limit of th at of the chiral Gaussian u nitary ens emble stud ied in the random matrix theory . Then Mark o vianit y is p r o ved for the three infinite- dimensional determinan tal pro cesses. Keyw ords Determinan tal pro cesses · Correlation k ernels · Random matrix theory · Infinite p article systems · Mark o v prop ert y · Enti re function and topology Mathematics Sub ject Classification ( 2010) 15B52 , 30C15, 47D07, 60G55 , 82C22 ∗ Department of Ph ysics, F aculty of Science and Engineering , Ch uo Universit y , Kasuga, Bunkyo-ku, T oky o 112-8 551, Japa n; e - mail: k ator i@phys.c huo-u.ac.jp † Department of Mathematics and Informatics, F aculty of Science, Chiba Universit y , 1-33 Y ay oi-cho, Inage-ku, Chiba 263-8 522, Japa n; e-mail: tanem ura@ math.s.chiba-u.ac.jp 1 1 In tro duc t ion Let M b e the space of nonnegative integer-v alued Rado n measures on R , whic h is a P ol- ish space with the vague top olo gy : let C 0 ( R ) b e the set of all con tinuous real-v alued functions with compact supp orts, and we say ξ n , n ∈ N conv erges to ξ v aguely , if lim n →∞ R R ϕ ( x ) ξ n ( dx ) = R R ϕ ( x ) ξ ( dx ) for an y ϕ ∈ C 0 ( R ). Eac h elemen t ξ of M can b e represen t ed as ξ ( · ) = P j ∈ Λ δ x j ( · ) with an index set Λ and a sequenc e of p oin ts x = ( x j ) j ∈ Λ in R satisfying ξ ( K ) = ♯ { x j : x j ∈ K } < ∞ for an y compact subset K ⊂ R . W e call an elemen t ξ of M an unlab eled configur a tion, and a sequence of p oints x a lab eled configuration. A pro b- abilit y measure o n a configuration space is called a determinan ta l p oin t pro cess o r F ermion p oin t pro cess, if its correlation functions a r e generally r epresen t ed by determinan ts [32, 33]. In the presen t pap er w e sa y that an M -v alued pro cess Ξ( t ) is determinantal , if the m ulti- time cor r elat io n functions for an y chos en series o f times are represen ted by determinan ts. In other w ords, a determinantal p r o c ess is an M -v alued pro cess suc h that, for any in teger M ∈ N = { 1 , 2 , . . . } , f = ( f 1 , f 2 , . . . , f M ) ∈ C 0 ( R ) M , a sequence of times t = ( t 1 , t 2 , . . . , t M ) with 0 < t 1 < · · · < t M < ∞ , if w e set χ t m ( x ) = e f m ( x ) − 1 , 1 ≤ m ≤ M , the momen t generating function of multitime distribution, Ψ t [ f ] ≡ E h exp n P M m =1 R R f m ( x )Ξ( t m , d x ) oi , is giv en by a F redholm determinan t Ψ t [ f ] = Det ( s,t ) ∈{ t 1 ,t 2 ,...,t M } 2 , ( x,y ) ∈ R 2 = Det h δ st δ ( x − y ) + K ( s, x ; t, y ) χ t ( y ) i , (1.1) with a lo cally in tegrable function K called a c orr elation kern e l [1 4, 16]. F inite- and infinite- dimensional determinan tal processes w ere in tro duced as m ulti- matrix models [7, 22], tiling mo dels [11], and surface g r o wth mo dels [30], and they ha v e b een extensiv ely studied. In the presen t pap er w e study t he infinite-dimensional determinan tal pro cesses desc ribing one- dimensional infinite particle systems with long-range repulsiv e interactions. They are ob- tained b y taking appro pria te N → ∞ limits of the N -particle system s of nonc ol liding diffu- sion pr o c esses , whic h dynamically simulate the eigen v alue statistics of the G aussian random matrix ensem bles studied in the random mat rix theory [2 1 , 8]. The purp ose of the presen t pap er is to prov e Markov pr op erty of the three infinite-dimensional determinan ta l pro cesses with the correlatio n k ernels called the extende d si ne, A iry, and Bessel kernels , whic h are rev ersible with r esp ect to the probability measures o bt a ined in the bulk scaling limit and the soft-edge scaling limit of the eigen v alue distribution in the Gaussian unitary ensem ble (GUE), a nd in t he hard-edge scaling limit o f tha t in the chir al Gaussian uni tary en semble (c hGUE), resp ectiv ely . Se e, for instance, [19] and the references therein. Dyson [6] intro duced a sto c hastic mo del of particles in R with a log-p oten tia l, wh ic h ob eys the sto c ha stic differen t ia l equations ( SD Es): dX j ( t ) = dB j ( t ) + X 1 ≤ k ≤ N ,k 6 = j dt X j ( t ) − X k ( t ) , 1 ≤ j ≤ N , t ∈ [0 , ∞ ) , (1.2) where B j ( t )’s are indep enden t one-dimensional standard Bro wnian motions. This is a sp e- cial case w ith the parameter β = 2 of Dyson’s Bro wnian motion models [6 , 21] but w e 2 will call it simply the Dyson model in this pap er. It is equiv alen t to a system of one- dimensional Bro wnian motions conditioned neve r to collide with each other [19]. F or the solution X j ( t ), j = 1 , 2 , . . . , N of (1.2) with initial v alues X j (0) = x j , j = 1 , 2 , . . . , N , w e put Ξ( t, · ) = P N j =1 δ X j ( t ) ( · ) and ξ N ( · ) = P N j =1 δ x j ( · ). The (unlab eled) Dyson mo del starting from ξ N is denoted by ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ N ). In the previous pa p er [16] w e show ed that f or any fixed unlab eled configuration ξ N , the pro cess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ N ) is determinan ta l with t he correlation k ernel K ξ N sp ecified by a n entire function Π 0 , whic h is explicitly give n by ( 2.1) b elo w in the presen t pap er. Then for configurations ξ ∈ M with ξ ( R ) = ∞ , a suffic ien t con- dition was give n so that the sequence of the pro cesses ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ ∩ [ − L,L ] ) con v erges to an M -v alued determinan tal pro cess as L → ∞ in the sense of finite dimensional distributions in the v ague top ology , where ξ ∩ [ − L, L ] denotes the restricted measure o f ξ o n an interv al [ − L, L ]. Note tha t ξ ∈ M implies ξ ∩ [ − L, L ]( R ) < ∞ for 0 < L < ∞ . The limit pro cess is the Dyson mo del with an infinite numb er of p articles and denoted b y ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ ). Recen tly , in [18] the tigh tness of t he sequence of pro cesses w as prov ed. The class of config- urations satisfying our condition, denoted b y Y , is lar g e enough to carry the Pois son p oin t pro cesses, G ibbs states with regular conditions, a s we ll as the determinan t a l (F ermion) p oin t pro cess µ sin with the sine kerne l: K sin ( x, y ) ≡ 1 2 π Z | k |≤ π dk e ik ( y − x ) = sin { π ( y − x ) } π ( y − x ) , x, y ∈ R , (1.3) where i = √ − 1. F rom the uniqueness of solutions of (1.2) , the Dyson mo del with a finite n umber of particles is a diffusion pro cess ( i. e . it is a strong Marko v pro cess ha ving a con tinuous path almost surely). Although the pro cess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ ) is g iven as the limit of a sequence of diffusion pro cesses ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ ∩ [ − L,L ] ), L ∈ N , the Marko v prop erty could b e lost in it. In the D yson mo del, b ecause of the long -range in t era ction, if the n um b er of particles is infinite, the probability distribution P ξ is not v aguely con tinuous with resp ect to the initial configuration ξ . There fore the Mark ovianit y of the pro cess is not readily concluded in the infinite-particle limit of a sequence of Mark ov pro cesses in the v ague top ology . F or a probability measure µ on M w e put P µ ( · ) = R M µ ( dξ ) P ξ ( · ). Supp ose t ha t ξ ( R ) = ∞ , µ -almost surely . F or proving Mark ov prop ert y of the pro cess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ ), it is sufficien t to find a subset b Y of M with a top ology T , whic h is stronger than the v ague top ology , and a sequence of pro ba bilit y measures { µ N } N ∈ N suc h that 1. P ξ ( · ) is con tin uous with resp ect to ξ in the top ology T , 2. P µ (Ξ( t ) ∈ b Y ) = 1, for any t ∈ [0 , ∞ ), 3. µ N ( { η ∈ b Y : η ( R ) = N } ) = 1, N ∈ N , 4. ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ N ) con verges ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ ) as N → ∞ in the sense of finite di- mensional distributions in the top ology T . 3 In this pa p er first w e will sho w that t hese conditions a re satisfied in the case that µ = µ sin , if w e use Y as b Y with the top ology T called the Φ 0 -mo der ate top olo gy defined b y (2.2) g iv en b elo w. W e also sho w that the pro cess ( { Ξ ( t ) } t ∈ [0 , ∞ ) , P µ sin ) is a version of the determinantal pro cess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ) asso ciated with the extende d sine kernel K sin with densit y 1: K sin ( s, x ; t, y ) ≡ 1 2 π Z | k |≤ π dk e k 2 ( t − s ) / 2+ ik ( y − x ) − 1 ( s > t ) p ( s − t, x | y ) =            Z 1 0 du e π 2 u 2 ( t − s ) / 2 cos { π u ( y − x ) } if t > s K sin ( x, y ) if t = s − Z ∞ 1 du e π 2 u 2 ( t − s ) / 2 cos { π u ( y − x ) } if t < s. (1.4) (Tw o pro cesses havin g the same state space are said to b e e quivale n t if they ha v e the same finite-dimensional distributions, and w e also sa y tha t eac h one is a version of other or they are v ersions of the same pro cess. See Section 1.1 o f [31].) Hence, w e conclude that the determinan tal pro cess with the extended sine k ernel is a Mark ov pro cess whic h is rev ersible with resp ect to t he determinan tal p o in t pro cess µ sin (Theorem 2.6). The ab o ve strategy will b e also used to show Mark ovianit y of the infinite-dimensional determinan tal pro cess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P J ν ) asso ciated with the extende d Bessel kern el K J ν : K J ν ( s, x ; t, y ) =                        Z 1 0 du e − 2 u ( s − t ) J ν (2 √ ux ) J ν (2 √ uy ) if s < t J ν (2 √ x ) √ y J ′ ν (2 √ y ) − √ xJ ′ ν (2 √ x ) J ν (2 √ y ) x − y if t = s − Z ∞ 1 du e − 2 u ( s − t ) J ν (2 √ ux ) J ν (2 √ uy ) if s > t, (1.5) x, y ∈ R + ≡ { x ∈ R : x ≥ 0 } , where J ν ( · ) is the Bessel function with index ν > − 1 defined b y J ν ( z ) = ∞ X n =0 ( − 1) n Γ( n + 1) Γ ( n + 1 + ν )  z 2  2 n + ν , z ∈ C with the gamma function Γ( x ) = R ∞ 0 e − u u x − 1 du (Theorem 2 .7). This pro cess is rev ersible with resp ect to t he determinan tal p o in t pro cess µ J ν with the Bessel k ernel K J ν ( x, y ) =    J ν (2 √ x ) √ y J ′ ν (2 √ y ) − √ xJ ′ ν (2 √ x ) J ν (2 √ y ) x − y if x 6 = y , ( J ν (2 √ x ) 2 − J ν +1 (2 √ x ) J ν − 1 (2 √ x ) if x = y , (1.6) with J ′ ν ( x ) = dJ ν ( x ) /dx . 4 Finally we will study the infinite-dimensional determinan ta l pro cess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P Ai ) asso ciated with the extende d Airy kernel K Ai : K Ai ( s, x ; t, y ) ≡        Z ∞ 0 du e − u ( t − s ) / 2 Ai( u + x )Ai( u + y ) if t ≥ s − Z 0 −∞ du e − u ( t − s ) / 2 Ai( u + x )Ai( u + y ) if t < s, (1.7) x, y ∈ R , where Ai( · ) is the Airy function defined b y Ai( z ) = 1 2 π Z R dk e i ( z k + k 3 / 3) , z ∈ C , whic h is reve rsible with resp ect to the determinan tal p oint pro cess µ Ai with the Airy k ernel K Ai ( x, y ) =    Ai( x )Ai ′ ( y ) − Ai ′ ( x )Ai( y ) x − y if x 6 = y (Ai ′ ( x )) 2 − x (Ai( x )) 2 if x = y , (1.8) where Ai ′ ( x ) = d Ai( x ) /dx . In this pro cess t he densit y ρ Ai ( x ) of particles is not b ounded and sho ws the same a symptotic behavior with the function b ρ ( 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 repulsiv e force from infinite num b er of pa rticles in the negativ e region x < 0 causes a p ositiv e drift with infinite strength. T o compensate it in o rder to obtain a stationary system, another negative drift with infinite strength should b e included in the pro cess. F or this reason we hav e to mo dify our strategy and w e intro duce not only a sequence of probability measures but also a sequence of appro ximate pro cesses with finite nu m b ers of particles ( { Ξ b ρ N ( t ) } t ∈ [0 , ∞ ) , P ξ N ) , N ∈ N , ha ving negativ e drifts suc h that their strength div erges as N go es to infinity . T o determine the N - dep endence of negativ e drifts , w e use the estimate (4.13) in Lemma 4.3. Then w e will sho w that t he limiting pro cess is a v ersion of ( { Ξ( t ) } t ∈ [0 , ∞ ) , P Ai ) and it is Mark ovian (Theorem 2.8). Remark 1. Other pro cesses hav ing infinite-dimensional determinan tal p oin t pro cesses as their statio nary measures ha v e b een constructed and Mark ovianit y o f systems w as prov ed in Boro din and Olshanski [3, 4, 5], Olshanski [24], and Bo ro din and Gorin [2]. They determined the state spaces and tra nsition functions asso ciated with F eller semi-gro ups and concluded that these infinite-dimensional determinan tal pro cesses are strong F eller pro cesses. In the presen t pap er, w e first prov e Mark ovianit y o f the determinan ta l pro cesse s with the extended sine k ernel K sin , the extended Airy ke rnel K Ai , and the extended Bessel kernel K J ν , whic h ha ve b een we ll-studied in the random matrix theory and its related fields. T o these pro cesses , the arg ument b y [3, 4, 5, 24, 2] can not b e applied, and the strong F eller prop erty ha s not y et prov ed. (See also Remark 2 give n at the end of Section 2.) The pap er is o rganized a s follows. In Section 2 preliminaries and main results are give n. In Section 3 the basic prop erties o f the correlation functions are summarized. Section 4 is dev oted to pro ofs o f results by using Lemma 4.3. In Section 5 Lemma 4.3 is prov ed by using the estimate of Hermite p olynomials giv en b y Planc herel and Rota c h [2 9]. 5 2 Preliminaries and Main Resu lts 2.1 Non-equilibrium dynamics F or ξ N ∈ M with ξ N ( R ) = N ∈ N and p ∈ N 0 ≡ N ∪ { 0 } w e consider the pro duct Π p ( ξ N , w ) = Y x ∈ supp ξ N G  w x , p  ξ ( { x } ) , w ∈ C , where G ( u, p ) =        1 − u, if p = 0 (1 − u ) exp  u + u 2 2 + · · · + u p p  , if p ∈ N . The functions G ( u, p ) are called the W eierstrass primary facto r s [20]. Then w e set Φ p ( ξ N , z , w ) ≡ Π p ( τ − z ξ N ∩ { 0 } c , w − z ) = Y x ∈ supp ξ N ∩{ z } c G  w − z x − z , p  ξ ( { x } ) , w , z ∈ C , where τ z ξ ( · ) ≡ P j ∈ Λ δ x j + z ( · ) for z ∈ C and ξ ( · ) = P j ∈ Λ δ x j ( · ). It w as pro v ed in Prop o sition 2.1 of [16] that the Dyson mo del ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ N ), starting from any fixed configuration ξ N ∈ M is determinantal with the correlation k ernel K ξ N giv en b y K ξ N ( s, x ; t, y ) = 1 2 π i Z R du I C iu ( ξ N ) dz p sin ( s, x | z ) Π 0 ( τ − z ξ N , iu − z ) iu − z p sin ( − t, iu | y ) − 1 ( s > t ) p sin ( s − t, x | y ) , (2.1) where C w ( ξ N ) denotes a closed contour on t he complex plane C encircling the p oin ts in supp ξ N on the r eal line R once in the p ositiv e direction but not the p oint w , and p sin ( t, x | y ) is the generalized heat k ernel: p sin ( t, x | y ) = 1 p 2 π | t | exp n − ( x − y ) 2 2 t o 1 ( t 6 = 0) + δ ( y − x ) 1 ( t = 0) , t ∈ R , x, y ∈ C . W e put M 0 = { ξ ∈ M : ξ ( x ) ≤ 1 for an y x ∈ R } . An y elemen t ξ ∈ M 0 has no multiple p oin ts and can b e iden tified with its supp ort whic h is a countable subs et of R . In case ξ N ∈ M 0 , (2.1) is rewritten a s K ξ N ( s, x ; t, y ) = Z R ξ N ( dx ′ ) Z R du p sin ( s, x | x ′ )Φ 0 ( ξ N , x ′ , iu ) p sin ( − t, iu | y ) − 1 ( s > t ) p sin ( s − t, x | y ) . F or ξ ∈ M , p ∈ N 0 , w , z ∈ C w e define Φ p ( ξ , z , w ) = lim L →∞ Φ p ( ξ ∩ [ − L, L ] , z , w ) 6 if the limit finitely exists. F or L > 0 , α > 0 and ξ ∈ M w e put M ( ξ , L ) = Z [ − L,L ] \{ 0 } ξ ( dx ) x , and M ( ξ ) = lim L →∞ M ( ξ , L ) , if the limit finitely exists, and for α > 0 w e put M α ( ξ ) =  Z { 0 } c 1 | x | α ξ ( dx )  1 /α . It is readily to see that Φ p ( ξ , z , w ) finitely exists and is not iden tically 0 if M p +1 ( ξ ) < ∞ , and that | Φ 0 ( ξ , z , w ) | < ∞ if | M ( ξ ) | < ∞ and M 2 ( ξ ) < ∞ . F or κ > 0, w e put g κ ( x ) = sgn( x ) | x | κ , x ∈ R , and η κ ( · ) = P ℓ ∈ Z δ g κ ( ℓ ) ( · ). F o r κ ∈ (1 / 2 , 1) and m ∈ N we denote b y Y κ,m the set of configuratio ns ξ satisfying the follo wing conditions (C.I) and (C.I I): (C.I) | M ( ξ ) | < ∞ , (C.I I) m ( ξ , κ ) ≡ max k ∈ Z ξ  [ g κ ( k ) , g κ ( k + 1)]  ≤ m. And w e put Y = [ κ ∈ (1 / 2 , 1) [ m ∈ N Y κ,m . Noting that the set { ξ ∈ M : m ( ξ , κ ) ≤ m } is relative ly compact with the v ague top olog y for eac h κ ∈ (1 / 2 , 1) and m ∈ N , we see that Y is lo cally compact. W e introduce the following top ology on Y . Definition 2.1 Supp ose that ξ , ξ n ∈ Y , n ∈ N . We say that ξ n c onver ges Φ 0 -mo der ately to ξ , if lim n →∞ Φ 0 ( ξ n , i, · ) = Φ 0 ( ξ , i, · ) uniformly on an y c omp act s e t of C . (2.2) It is easy to see that (2.2) is satisfied, if ξ n con ve rges to ξ v aguely and the fo llo wing tw o conditions hold: lim L →∞ sup n> 0      M ( ξ n ) − M ( ξ n , L )      = 0 , (2.3) lim L →∞ sup n> 0      M 2 ( ξ n ∩ [ − L, L ] c )      = 0 . (2.4) Note that for any a ∈ R and z ∈ C lim n →∞ Φ 0 ( ξ n , a, z ) = Φ 0 ( ξ , a, z ), if ξ n con ve rges Φ 0 - mo derately to ξ and a / ∈ supp ξ . W e denote the space of M -v alued con tin uous functions defined on [0 , ∞ ) by C ([0 , ∞ ) → M ). W e hav e o btained the fo llowing results. (See Theorem 2.4 of [16] and Theorem 1.4 of [18].) 7 Prop osition 2.2 (i) If ξ ∈ Y , the pr o c ess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ ∩ [ − L,L ] ) c onver ges to the d e - terminantal p r o c ess with a c orr elation kernel K ξ as L → ∞ , w e akly on the p ath sp ac e C ([0 , ∞ ) → M ) . In p articular, w hen ξ ∈ Y 0 ≡ Y ∩ M 0 , K ξ is given by K ξ ( s, x ; t, y ) = Z R ξ ( dx ′ ) Z R du p sin ( s, x | x ′ )Φ 0 ( ξ , x ′ , iu ) p sin ( − t, iu | y ) − 1 ( s > t ) p sin ( s − t, x | y ) . (ii) Supp ose that ξ , ξ n ∈ Y κ m , n ∈ N , for some κ ∈ (1 / 2 , 1) a n d m ∈ N . If ξ n c on- ver ges Φ 0 -mo der ately to ξ , then the pr o c ess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ n ) c onver ges to the pr o c ess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P ξ ) as n → ∞ , we akly on the p ath sp ac e C ([0 , ∞ ) → M ) . Let M + = { ξ ∩ R + : ξ ∈ M } . W e consider a one-parameter family of M + -v alued pro cesses with a para meter ν > − 1, Ξ ( ν ) ( t, · ) = X j ∈ Λ δ X ( ν ) j ( t ) ( · ) , t ∈ [0 , ∞ ) , (2.5) where X ( ν ) j ( t )’s satisfy the SDEs, dX ( ν ) j ( t ) = 2 q X ( ν ) j ( t ) dB j ( t ) + 2( ν + 1) dt + 4 X ( ν ) j ( t ) X k : k 6 = j 1 X ( ν ) j ( t ) − X ( ν ) k ( t ) dt, j ∈ Λ , t ∈ [0 , ∞ ) (2.6) and if − 1 < ν < 0 with a reflection w all at the origin. F or a giv en configuration ξ N ∈ M + of finite particles, ξ N ( R + ) = N ∈ N , the pro cess starting from ξ N is denoted b y ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P ξ N ) a nd called the n o nc ol liding squar e d Bessel pr o c esses with ind ex ν . In Theorem 2 .1 of [17] it was pro ve d that ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P ξ N ) is determinan tal with the cor- relation k ernel K ξ N ν ( s, x ; t, y ) = 1 2 π i Z 0 −∞ du I C u ( ξ N ) dz p ( ν ) ( s, x | z ) Π 0 ( τ − z ξ N , u − z ) u − z p ( ν ) ( − t, u | y ) − 1 ( s > t ) p ( ν ) ( s − t, x | y ) , (2.7) where for t ∈ R and x, y ∈ C p ( ν ) ( t, y | x ) = 1 2 | t |  y x  ν / 2 exp  − x + y 2 t  I ν  √ xy | t |  1 ( t 6 = 0 , x 6 = 0) + y ν (2 | t | ) ν +1 Γ( ν + 1) exp  − y 2 t  1 ( t 6 = 0 , x = 0) + δ ( y − x ) 1 ( t = 0) . (2.8) Note that f or t ≥ 0 and x, y ∈ R + , p ( ν ) ( t, y | x ) is the transition densit y function of 2( ν + 1)- dimensional squared Bessel pro cess. In case ξ N ∈ M + ∩ M 0 , (2.7) is equal to K ξ N ν ( s, x ; t, y ) = Z ∞ 0 ξ N ( dx ′ ) Z 0 −∞ du p ( ν ) ( s, x | x ′ )Φ 0 ( ξ N , x ′ , u ) p ( ν ) ( − t, u | y ) − 1 ( s > t ) p ( ν ) ( s − t, x | y ) 8 F or κ ∈ (1 , 2) and m ∈ N , let Y + κ,m b e the set of configurations ξ of M + satisfying ( C.I I) , and put Y + = [ κ ∈ (1 , 2) [ m ∈ N Y + κ,m . Note that Y + is lo cally compact in the v ague top olo g y . The fo llo wing prop osition is a slight mo dification of the results stated in Section 2 of [17], whic h can b e pro ve d b y the same pro cedure g iven there. Prop osition 2.3 (i) If ξ ∈ Y + , the pr o c ess ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P ξ ∩ [ − L,L ] ) c onv e r ges to the de- terminantal pr o c ess with a c orr elation kernel K ξ ν as L → ∞ in the sense of finite dim ensional distributions. In p articular, w hen ξ ∈ Y + 0 ≡ Y + ∩ M 0 , K ξ ν is given by K ξ ν ( s, x ; t, y ) = Z ∞ 0 ξ ( dx ′ ) Z 0 −∞ du p ( ν ) ( s, x | x ′ )Φ 0 ( ξ N , x ′ , u ) p ( ν ) ( − t, u | y ) − 1 ( s > t ) p ( ν ) ( s − t, x | y ) (ii) Supp ose that ξ , ξ n ∈ Y + κ,m , n ∈ N , f o r some κ ∈ (1 , 2) and m ∈ N . If ξ n c on- ver ges Φ 0 -mo der ately to ξ , then the pr o c ess ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P ξ n ) c onver ges to the pr o c ess ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P ξ ) as n → ∞ in the sense of finite dimensiona l distributions. Let b ρ N , N ∈ N b e a sequence of nonnegativ e functions on R suc h that b ρ N ( x ) = 0 for x ≥ 0, R R dx b ρ N ( x ) = N and b ρ N ( x ) ր b ρ ( x ) = ( √ − x/π ) 1 ( x < 0 ) , N → ∞ . F or ξ N ∈ M with ξ N ( R ) = N w e put M b ρ N ( ξ N ) = Z { 0 } c b ρ N ( x ) dx − ξ N ( dx ) x . and define Φ b ρ N ( ξ N , w ) ≡ exp " w M b ρ N ( ξ N ) # Π 1 ( ξ N ∩ { 0 } c , w ) , z ∈ C , Φ b ρ N ( ξ N , z , w ) ≡ Φ b ρ N ( τ − z ξ N , w − z ) , w , z ∈ C . F or ξ ∈ M with ξ ( R ) = ∞ we put M A ( ξ ) = lim N →∞ M b ρ N ( ξ N ) , if ξ N con ve rges to ξ , N → ∞ , with the v ague top ology and lim L →∞ sup N ∈ N Z | x | >L b ρ N ( x ) dx − ξ N ( dx ) x = 0 , (2.9) is satisfied. Then M A ( ξ ) is also represen ted a s M A ( ξ ) = lim L →∞ Z 0 < | x | t ) q ( t, s, x − y ) , (2.12) where q ( s, t, y − x ) , s, t ∈ R , s 6 = t, x, y ∈ C is given b y 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  . Note tha t q ( s, t, y − x ), t > s ≥ 0, x, y ∈ R is the t ransition density function of the pro cess B ( t ) + t 2 / 4, where B ( t ) , t ∈ [0 , ∞ ) is the one-dimensional standard Brownian motion. In case ξ N ∈ M 0 , (2.12) is equal to K ξ N b ρ N ( s, x ; t, y ) = Z R ξ ( dx ′ ) Z R du q (0 , s, x − x ′ )Φ b ρ N ( ξ N , x ′ , iu ) q ( t, 0 , iu − y ) − 1 ( s > t ) q ( t, s, x − y ) . 10 F or κ ∈ (1 / 2 , 2 / 3) and m ∈ N , w e denote b y Y A κ,m the set of configurations ξ satisfying the conditions ( C.I- A ) | M A ( ξ ) | < ∞ , and ( C.I I ). And w e define the space of configurations Y A = [ κ ∈ (1 / 2 , 2 / 3) [ m ∈ N Y A κ,m . Note that Y A is lo cally compact. W e introduce the following top ology on Y A . Definition 2.4 Supp ose that ξ , ξ n ∈ Y A , n ∈ N . We say that ξ n c onver ges Φ A -mo der ately to ξ , if lim n →∞ Φ A ( ξ n , i, · ) = Φ A ( ξ , i, · ) uniformly on an y c omp act s e t of C . (2.13) The fo llo wing prop osition is a sligh t mo dification of the results stated in Section 2.4 o f [15]. Prop osition 2.5 L et ξ ∈ Y A and ξ N , N ∈ N b e a se quenc e of c onfigur ations such that ξ N ( R ) = N and ξ N c onver ges to ξ with the vag ue top olo gy. Supp ose that (2.9 ) is satisfie d and max N ∈ N m ( ξ N , κ ) ≤ m , for s o me κ ∈ ( 1 / 2 , 2 / 3) , m ∈ N . (i) The family of the distributions of the pr o c e sses ( { Ξ b ρ N ( t ) } t ∈ [0 , ∞ ) , P ξ N ) is tight in the sp ac e of pr ob ability me asur es on C ([0 , ∞ ) → M ) . (ii) The se q uenc e of the pr o c esses ( { Ξ b ρ N ( t ) } t ∈ [0 , ∞ ) , P ξ N ) c onver ges to the de termi n antal pr o c ess ( { Ξ A ( t ) } t ∈ [0 , ∞ ) , P ξ ) with a c orr elation kernel K ξ A as N → ∞ , we akly on the p ath sp ac e C ([0 , ∞ ) → M ) . In p articular, when ξ ∈ Y A 0 ≡ Y A ∩ M 0 , K ξ A is given by K ξ A ( s, x ; t, y ) = Z R ξ ( dx ′ ) Z R du q (0 , s, x − x ′ )Φ A ( ξ , x ′ , iu ) q ( t, 0 , iu − y ) − 1 ( s > t ) q ( t, s, x − y ) . (iii) Supp ose that ξ , ξ n ∈ Y A κ,m , n ∈ N , fo r some κ ∈ (1 / 2 , 2 / 3) and m ∈ N . If ξ n c on- ver ges Φ A -mo der ately to ξ , then the pr o c ess ( { Ξ A ( t ) } t ∈ [0 , ∞ ) , P ξ n ) c onve r ges to the pr o c ess ( { Ξ A ( t ) } t ∈ [0 , ∞ ) , P ξ ) as n → ∞ we akly on the p ath sp ac e C ([0 , ∞ ) → M ) . By the same argumen t in [18] we can obtain the first assertion (i), under which the ot her assertions (ii) a nd (iii) can b e prov ed b y the same pro cedure give n in Section 2 .4 o f [15]. Hence, w e skip the pro of of this prop osition. 2.2 Main results F or ξ ∈ Y w e put T t f ( ξ ) = E ξ h f (Ξ( t )) i , t ≥ 0 , (2.14) 11 for a b ounded v a g uely con tin uous function f on M , where E ξ represen ts the exp ecta- tion with resp ect to the probability measure P ξ . When ξ , ξ n ∈ Y κ,m , n ∈ N , for some κ ∈ (1 / 2 , 1) and m ∈ N , T t f ( ξ n ) con v erges to T t f ( ξ ), if ξ n con ve rges Φ 0 -mo derately to ξ , as n → ∞ . W e denote by L 2 ( M , µ ) the space of square integrable functions on M with resp ect t o the probabilit y measure µ , whic h is equipp ed with the inner pro duct h f , g i µ ≡ Z M µ ( dξ ) f ( ξ ) g ( ξ ) , f , g ∈ L 2 ( M , µ ). W e write the exp ectation with resp ect to the probabilit y measure P sin as E sin . The first main theorem o f the presen t pap er is the follow ing. Theorem 2.6 (i) µ sin ( Y ) = 1 an d T t is extende d to the c ontr action op er ator on L 2 ( M , µ sin ) . (ii) T he pr o c esses ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ sin ) an d ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ) ar e version s of the same de- terminantal pr o c ess. In p articular, for any t ≥ 0 E sin h f 0 (Ξ(0)) f 1 (Ξ( t )) i = h f 0 , T t f 1 i µ sin , f 0 , f 1 ∈ L 2 ( M , µ sin ) . (2.15) (iii) The r eversible pr o c ess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ) is Markov ian. F or ξ ∈ Y + w e put T ( ν ) t f ( ξ ) = E ξ h f (Ξ ( ν ) ( t )) i , t ≥ 0 , (2.16) for a b ounded contin uous function f on M + . When ξ , ξ n ∈ Y + κ,m , n ∈ N , for some κ ∈ (1 , 2) and m ∈ N , T ( ν ) t f ( ξ n ) con ve rges to T ( ν ) t f ( ξ ), if ξ n con ve rges Φ 0 -mo derately to ξ , as n → ∞ . W e write the exp ectation with resp ect to the pro babilit y measure P J ν as E J ν . The second main theorem of the presen t pap er is t he following. Theorem 2.7 (i) µ J ν ( Y + ) = 1 a nd T ( ν ) t is extende d to the c ontr action op er ator o n L 2 ( M + , µ J ν ) . (ii) The pr o c esses ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P µ J ν ) and ( { Ξ( t ) } t ∈ [0 , ∞ ) , P J ν ) ar e versions of the same determinantal pr o c ess. In p articular, for a n y t ≥ 0 E J ν h f 0 (Ξ(0)) f 1 (Ξ( t )) i = h f 0 , T ( ν ) t f 1 i µ J ν , f 0 , f 1 ∈ L 2 ( M + , µ J ν ) . (2.17 ) (iii) The r eversible pr o c ess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P J ν ) is Markovian. F or ξ ∈ Y A w e put T A t f ( ξ ) = E ξ h f (Ξ A ( t )) i , t ≥ 0 , (2.18) for a b ounded con tin uous function f on M . When ξ , ξ n ∈ Y A κ,m , n ∈ N , fo r some κ ∈ (1 / 2 , 2 / 3) and m ∈ N , T A t f ( ξ n ) conv erges to T A t f ( ξ ), if ξ n con ve rges Φ A -mo derately to ξ , as n → ∞ . W e write the exp ectation with resp ect to the probabilit y measure P Ai as E Ai . The last main theorem o f the presen t pap er is the following. Theorem 2.8 (i) µ Ai ( Y A ) = 1 a nd T A t is extende d to the c ontr action op er ator o n L 2 ( M , µ Ai ) . 12 (ii) Th e pr o c esses ( { Ξ A ( t ) } t ∈ [0 , ∞ ) , P µ Ai ) and ( { Ξ( t ) } t ∈ [0 , ∞ ) , P Ai ) ar e vers i o ns of the same determinantal pr o c ess. In p articular, for a n y t ≥ 0 E Ai h f 0 (Ξ(0)) f 1 (Ξ( t )) i = h f 0 , T A t f 1 i µ Ai , f 0 , f 1 ∈ L 2 ( M , µ Ai ) . (2.19 ) (iii) The r eversible pr o c ess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P Ai ) is Markovian. Remark 2. A function f on the configuration space M is said to b e p olynomial, if it is written in the for m f ( ξ ) = F R R φ 1 ( x ) ξ ( dx ) , R R φ 2 ( x ) ξ ( dx ) , . . . , R R φ k ( x ) ξ ( dx )  with a p oly- nomial function F on R k , k ∈ N , and smo oth functions φ j , 1 ≤ j ≤ k on R with compact supp orts. Let ℘ b e the set of all p o lynomial functions on M . In [14 ] we in tr o duced a class of determinantal rev ersible pro cesses including the three pro cesses ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ), ( { Ξ( t ) } t ∈ [0 , ∞ ) , P J ν ) and ( { Ξ( t ) } t ∈ [0 , ∞ ) , P Ai ), and show ed in Prop osition 7.2 that for any el- emen t ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ ) of the class with the rev ersible measure µ , the following relation holds: lim t → 0 − 1 t E µ  f (Ξ(0)) g (Ξ( t ))  = Z R ρ ( x ) dx Z M µ x ( dη ) ∂ ∂ x f ( η + δ x ) ∂ ∂ x g ( η + δ x ) ≡ E µ 0 ( f , g ) , for an y f , g ∈ ℘, where ρ is the one-p oin t corr elat io n function of µ and µ x is the Palm m e asur e of µ , satisfying the follow ing relation with µ , Z M µ ( dξ ) ξ ( K ) f ( ξ ) = Z K dxρ ( x ) Z M µ x ( dη ) f ( η + δ x ) for an y f ∈ ℘ a nd an y compact subset K of R . Theorems 2.6, 2.7 and 2.8 imply that the bilinear forms ( E µ sin 0 , ℘ ), ( E µ J ν 0 , ℘ ) and ( E µ Ai 0 , ℘ ) are pre-Diric hlet forms [10] for t he three pro cess ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ), ( { Ξ( t ) } t ∈ [0 , ∞ ) , P J ν ) and ( { Ξ( t ) } t ∈ [0 , ∞ ) , P Ai ), resp ectiv ely . Sp ohn [34] considered a n infinite particle system obtained by taking the N → ∞ limit of (1.2) and studied the equilibrium dynamics with resp ect to the determinan t al p oint pro cess µ sin . F r o m the Dirichlet form approac h Osada [27] constructed the infinite particle systems represen ted b y the diffusion pro cess asso ciated with the Dirichlet form whic h is the minimal closed extension of ( E µ sin 0 , ℘ ). Recen tly he pro v ed that this system satisfies the SD Es (1 .2) with N = ∞ [28]. The equiv a lence of the determinan ta l pro cesses ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ) and the infinite-dimensional equilibrium dynamics of Sp ohn and Osada is, how ev er, not ye t pro ved. It is relev an t to the uniqueness of Mark ov extensions of the pre-Diric hlet form a sso ciated with an infinite particle system with long- range in t eraction, whic h is an in teresting op en problem. F o r an infinite particle syste m with finite-rang e in t era ctio n, there are some results on the uniqueness pro blem (see, e.g., [3 5 ]). 13 3 Correlatio n function s 3.1 Some estimates for determinan tal p oin t pro cesses Let µ b e a probabilit y measure on M with correlation functions ρ m ( x m ), x m ∈ R m , m ∈ N . Then for f ∈ C 0 ( R ) and θ ∈ R Ψ( f , θ ) ≡ Z M µ ( dξ ) e θ R R d f ( x ) ξ ( dx ) = ∞ X m =0 1 m ! Z R m m Y j =1 dx j m Y j =1  e θ f ( x j ) − 1  ρ m  x m  . Let K b e a symmetric linear op erator with k ernel K . In this section w e assume that t he op- erator K satisfies Condition A in [32]. The probability measure µ is called a determinan ta l p oin t pro cess with correlation kerne l K , if its correlation functions are represen ted by ρ m ( x m ) = det 1 ≤ j,k ≤ m h K ( x j , x k ) i . W e often write ρ f o r ρ 1 , a nd ρ ( A ) f or R A ρ ( x ) dx , A ∈ B ( R ), resp ectiv ely . In this subsection w e giv e some lemmas whic h will b e used in Section 4. Lemma 3.1 L et µ b e a determinantal p oint pr o c ess. T hen for any b ounde d close d interval D of R , we have Z M µ ( dη )    η ( D ) − Z D ρ ( x ) dx    2 k ≤ (3 ρ ( D )) k , k ∈ N . (3.1) Pr o of. Let K b e the correlation ke rnel for the determinan tal p o in t pro cess µ . W e put K D ( x, y ) = 1 D ( x ) K ( x, y ) 1 D ( y ) with 1 D ( x ) ≡ 1 ( x ∈ D ) and in tro duce the linear op erator K D defined b y K D f ( x ) = Z R K D ( x, y ) f ( y ) dy , f ∈ L 2 ( R , d x ) . Since D is compact, K D is in the trace class. Then nonzero sp ectrums of the op erator are eigen v alues { λ j } j ∈ Λ with 0 < λ j ≤ 1, j ∈ Λ. Hence, the distribution of η ( D ) under µ is identical with the distribution of the random v ariable P j ∈ Λ X j , where X j , j ∈ Λ are indep enden t Bernoulli random v ariables with P ( X j = 1) = λ j and P ( X j = 0) = 1 − λ j (see for instance [32]) . In fact w e can readily see that Z M e θ η ( D ) µ ( dη ) = Det  δ ( x − y ) + K ( x, y )( e θ 1 D ( x ) − 1)  = Det  δ ( x − y ) + K D ( x, y )( e θ − 1)  = Y j ∈ Λ  1 + λ j ( e θ − 1)  , whic h coincides with the generating function of P j ∈ Λ X j . By t his iden tity of distributions w e ha v e b Ψ( θ ) ≡ Z M µ ( dη ) exp  θ  η ( D ) − Z M µ ( dη ) η ( D )  2  = E " exp ( θ X j ∈ Λ ( X j − E X j ) 2 )# = Y j ∈ Λ e θ λ 2  1 − λ j (1 − e − 2 θλ j )  . 14 By simple calculations with the relation 0 ≤ λ j ≤ 1, we see that, for any k ∈ N d k dθ k e θ λ 2 j ≤ d k dθ k e θ λ j , d k dθ k { 1 − λ j + λ j e − 2 θλ j } ≤ d k dθ k e 2 θλ j , θ ≥ 0 . Hence, Z M µ ( dη )    η ( D ) − Z M µ ( dη ) η ( D )    2 k = d k dθ k b Ψ( θ )    θ = 0 ≤ d k dθ k exp ( 3 θ X j ∈ Λ λ j )      θ = 0 ≤ 3 X j ∈ Λ λ j ! k . Since R D ρ ( x ) dx = R M µ ( dξ ) η ( D ) = E h P j ∈ Λ X j i = P j ∈ Λ λ j , w e obtain the lemma. Lemma 3.2 L et ξ ∈ M and ρ b e the nonne gative function on R . Supp ose that ther e exist ε ∈ (0 , 1) , C 1 > 0 and L 1 > 0 such that     ξ ([0 , L ]) − Z L 0 ρ ( x ) dx     ≤ C 1 L ε ,     ξ ([ − L, 0 )) − Z 0 − L ρ ( x ) dx     ≤ C 1 L ε , L ≥ L 1 . (3.2) (i) If lim L →∞ Z 1 ≤| x |≤ L ρ ( x ) dx x finitely exits , (3.3) then ξ sa tisfi e s ( C.I) . (ii) If a nonne gative function ρ on R satisfies     Z | x |≥ L ρ ( x ) dx − ρ ( x ) dx x     ≤ C 2 L − δ , (3.4) for some δ > 0 , then ξ satisfies     Z | x |≥ L ρ ( x ) dx − ξ ( dx ) x     ≤ 3 C 1 1 − ε L ε − 1 + C 2 L − δ . (3.5) Pr o of. By using the in tegration by parts formula with (3.2), we see that for L 2 > L ≥ L 1     Z L 2 L ρ ( x ) dx x − Z L 2 L ξ ( dx ) x     ≤ 2 C 1 L ε − 1 + C 1 Z L 2 L x ε − 2 dx ≤ C 1 1 − ε (3 L ε − 1 − L 2 ε − 1 ) . Similarly , we hav e     Z − L − L 2 ρ ( x ) dx x − Z − L − L 2 ξ ( dx ) x     ≤ C 1 1 − ε (3 L ε − 1 − L 2 ε − 1 ) . Then for L ≥ L 1     Z | x |≥ L ξ ( dx ) x − Z | x |≥ L ρ ( x ) dx x     ≤ 3 C 1 1 − ε L ε − 1 . Then (i) and (ii) are deriv ed easily . 15 Lemma 3.3 L et µ b e a pr ob ability me asur e on M with c orr elation functions ρ m , m ∈ N . Supp ose that ther e exist m ′ ∈ N and p < m ′ − 1 such that Z M µ ( dξ )    ξ ([0 , L )) − Z L 0 ρ ( x ) dx    m ′ = O ( L p ) , L → ∞ , (3.6) Z M µ ( dξ )    ξ ([ − L, 0 )) − Z 0 − L ρ ( x ) dx    m ′ = O ( L p ) , L → ∞ . (3.7) (i) In c ase (3.3 ) holds and X k ∈ Z Z [ g κ ( k ) ,g κ ( k +1)] m ρ m ( x m ) d x m < ∞ (3.8) is satisfie d for s o me m ∈ N and κ ∈ (1 / 2 , 1) , then µ ( Y ) = 1 . (ii) In c ase µ ( M + ) = 1 and (3.8) is satisfie d for some m ∈ N and κ ∈ (1 , 2) , then µ ( Y + ) = 1 . (iii) I n c ase (3.4) holds with ρ = b ρ and (3 .8) s a tisfie d for some m ∈ N and κ ∈ ( 1 / 2 , 2 / 3) , then µ ( Y A ) = 1 . Pr o of. T a k e ε ∈ (( p + 1) /m ′ , 1) . Using Cheb yshev’s inequality with (3.6) and (3.7 ) , we can find a p ositiv e constant C suc h that µ  | ξ ((0 , L ]) − Z L 0 ρ ( x ) dx | ≥ C L ε  ≤ C L p − m ′ ε , and µ  | ξ ([ − L, 0)) − Z 0 − L ρ ( x ) dx | ≥ C L ε  ≤ C L p − m ′ ε Since p − m ′ ε < − 1, w e hav e X L ∈ N µ  | ξ ((0 , L ]) − Z L 0 ρ ( x ) dx | ) | ≥ C L ε  < ∞ , X L ∈ N µ  | ξ ([ − L, 0)) − Z 0 − L ρ ( x ) dx | ≥ C L ε  < ∞ , and so the condition (3.2) is satisfied µ - a.s. ξ by the Bo rel-Can telli lemma. Sinc e w e ha v e assumed the condition (3 .3 ) in (i), w e can apply Lemma 3.2 and conclude that ( C.I ) ho lds for µ - a.s. ξ . Similarly , since the condition (3.4 ) with ρ = b ρ is provided in (iii), we conclude that ( C.I- A ) ho lds for µ -a.s. ξ . The condition (3 .8) implies X k ∈ Z µ  ξ ( g κ ( k ) , g κ ( k + 1)) > m  < ∞ . Then, b y the Borel-Can t elli lemma again, ( C.I I ) is deriv ed. Therefore, we obtain the lemma. 16 3.2 Equilibrium dynamics The Dyson mo del starting fro m N p oin t s all at the origin is determinan ta l with the corre- lation ke rnel K N ( s, x ; t, y ) =            1 √ 2 s N − 1 X k =0  t s  k / 2 ϕ k  x √ 2 s  ϕ k  y √ 2 t  if s ≤ t − 1 √ 2 s ∞ X k = N  t s  k / 2 ϕ k  x √ 2 s  ϕ k  y √ 2 t  if s > t, (3.9) where h k = √ π 2 k k ! a nd ϕ k ( x ) = 1 √ h k e − x 2 / 2 H k ( x ) , (3.10) with the Hermite p olynomials H k , k ∈ N 0 (see Eynard a nd Meh ta [7]). The distribution of the pro cess at time t > 0 is iden tical with µ GUE N ,t , the distribution of eigen v alues in the G UE with v ariance t . The extended sine kerne l K sin with densit y 1 is deriv ed a s the bulk scaling limit [22]: K N  2 N π 2 + s, x ; 2 N π 2 + t, y  → K sin ( s, x ; t, y ) , N → ∞ , (3.11) whic h implies ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ GUE N, 2 N /π 2 ) f . d . − → ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ) , N → ∞ , in the v ague top ology , (3.12) where f . d . − → means the conv ergence in the sense of finite dimensional distributions. The noncolliding squared Bessel pro cess star t ing from N p oints all at t he origin is deter- minan tal with the correlation k ernel K ( ν ) N ( s, x ; t, y ) =            1 2 s N − 1 X k =0  t s  k ϕ ν k  x 2 s  ϕ ν k  y 2 t  , if s ≤ t , − 1 2 s ∞ X k = N  t s  k ϕ ν k  x 2 s  ϕ ν k  y 2 t  , if s > t , where ϕ ν k ( x ) = p Γ( k + 1) / Γ( ν + k + 1) x ν / 2 L ν k ( x ) e − x/ 2 with the Laguerre p olynomials L ν k ( x ) , k ∈ N 0 with parameter ν > − 1 [13]. Let µ ( ν ) N ,t b e t he distribution of the pro cess a t time t > 0. In the case t ha t ν ∈ N 0 , the probability measure µ ( ν ) N ,t is iden tical with the distribution of eigen v alues in the c hGUE with v ariance t . The extended Bessel ke rnel K J ν is deriv ed as the hard edge scaling limit [9, 36]: K ( ν ) N ( N + s, x ; N + t, y ) → K J ν ( s, x ; t, y ) . N → ∞ , (3.13) 17 whic h implies ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P µ ( ν ) N,N ) f . d . − → ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P ( ν ) ) , N → ∞ , in the v ague top olog y . (3.14) The extended Airy k ernel is deriv ed as the soft-edge scaling limit [30, 1 2, 23]: K N  N 1 / 3 + s, 2 N 2 / 3 + N 1 / 3 s − s 2 4 + x ; N 1 / 3 + t, 2 N 2 / 3 + N 1 / 3 t − t 2 4 + y  → K Ai ( s, x ; t, y ) , N → ∞ . (3.15) W e in tro duce the nonnegativ e function f sc ( x ) = (2 /π ) √ 1 − x 2 1 ( | x | ≤ 1), and put ρ N sc ( t, x ) = r N 4 t f sc  x 2 √ N t  = 1 2 π t √ 4 tN − x 2 1 ( | x | ≤ 2 √ tN ) , asso ciated with Wigner’s semicircle law of eigen v alue distribution in the GUE [21]. Then w e put b ρ N sc ( x ) = ρ N sc ( N 1 / 3 , 2 N 2 / 3 + x ) = N 1 / 3 2 f sc  1 + x 2 N 2 / 3  1 ( x < 0) = 1 π r − x  1 + x 4 N 2 / 3  1 ( x < 0) . (3.16) Here w e note that b ρ N sc ( x ) ր b ρ ( x ), N → ∞ and Z R dx b ρ N sc ( x ) = N , and Z 0 − 4 N 2 / 3 dx b ρ N sc ( x ) − x = N 1 / 3 . Then (3.15) implies ( { Ξ b ρ N sc ( t ) } t ∈ [0 , ∞ ) , P τ − 2 N 2 / 3 µ GUE N,N 1 / 3 ) f . d . − → ( { Ξ( t ) } t ∈ [0 , ∞ ) , P Ai ) , N → ∞ , in the v ague top olog y . (3.17) 4 Pro ofs of Results 4.1 Pro of of (i) of Theorems 2.6 and 2.7 T o obtain (i) o f the theorems it is enough to sho w µ sin ( Y ) = 1 and µ J ν ( Y + ) = 1, since the other claims are deriv ed from the facts that the op erator T t and T ( ν ) t are of contraction and the set of a ll b ounded con tin uous f unctions on M is dense in L 2 ( M , µ sin ) and L 2 ( M , µ J ν ). First w e consider the probabilit y measure µ sin . Since its dens it y ρ sin is constan t, the condition (3.3) is satisfied for ρ = ρ sin . Note tha t any correlation ρ m is b ounded, b ecause the k ernel K sin is b ounded. Then if w e take κ ∈ (1 / 2 , 1) a nd m ∈ N satisfying (1 − κ ) m > 1, 18 w e see that the condition (3.8) is satisfied for ρ = ρ sin . F r om Lemma 3.1 with k = 2, we see that the conditions (3.6 ) and (3.7 ) a re satisfied with m ′ = 4 , p = 2 and ρ = ρ sin . Th us, µ sin ( Y ) = 1. F or µ J ν , its density ρ ( ν ) is b ounded a nd satisfies R ∞ 0 { ρ ( ν ) ( x ) /x } dx < ∞ . Hence, b y the same arg ument as ab ov e w e obtain µ J ν ( Y + ) = 1. Remark 3. When µ λ is the Poisson po in t pro cess with a n in tensit y measure λdx , λ > 0, ρ m ( x m ) = λ m . Then w e 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 µ λ ( Y ) = 1. W e can also sho w that measures suc h as Gibbs states with regular conditions are a pplicable to Lemma 3.3 . 4.2 Pro of of (ii) and (iii) of T h eorems 2.6 and 2.7 The follo wing lemma states stronger prop erties than (3.12) and (3.14) and is the k ey to pro ving (ii) and (iii) of Theorems 2 .6 and 2 .7. The pro o f of this lemma is given in the next subsection. Lemma 4.1 (i) F or any 0 ≡ t 0 < t 1 < t 2 < · · · < t M < ∞ , M ∈ N 0 , a n d b ounde d functions g j , 0 ≤ j ≤ M , w hich is Φ 0 -mo der ately c ontinuous on Y m,κ for any m ∈ N and κ ∈ (1 / 2 , 1) , lim N →∞ E µ GUE N, 2 N /π 2 " M Y j =0 g j (Ξ( t j )) # = E sin " M Y j =0 g j (Ξ( t j )) # . (4.1) In p articular, ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ GUE N, 2 N /π 2 ) f . d . − → ( { Ξ( t ) } t ∈ [0 , ∞ ) , P sin ) , N → ∞ , in the sense of finite dimens i o nal distributions in the Φ 0 -mo der ate top olo gy. (ii) F or an y 0 ≡ t 0 < t 1 < t 2 < · · · < t M < ∞ , M ∈ N 0 , an d b ounde d functions g j , 0 ≤ j ≤ M , which is Φ 0 -mo der ately c ontinuous o n Y + m,κ for any m ∈ N a nd κ ∈ (1 , 2) , lim N →∞ E µ ( ν ) N,N " M Y j =0 g j (Ξ( t j )) # = E J ν " M Y j =0 g j (Ξ( t j )) # . (4.2) In p articular, ( { Ξ ( ν ) ( t ) } t ∈ [0 , ∞ ) , P µ ( ν ) N,N ) f . d . − → ( { Ξ( t ) } t ∈ [0 , ∞ ) , P J ν ) , N → ∞ , in the sense of fin ite dimensional distributions in the Φ 0 -mo der ate top olo gy. Pr o of o f (ii) and (iii) in T h e or ems 2.6 and 2.7. W e first give the pro of for Theorem 2.6. Let f j , 0 ≤ j ≤ M b e b ounded v aguely contin uous functions on M and 0 = t 0 ≤ t 1 < t 2 < · · · < t M < ∞ , M ∈ N . Theorem 2.6 (ii) is concluded from the equalit y Z M µ sin ( dξ ) E ξ " M Y j =0 f j (Ξ( t j )) # = lim N →∞ Z M µ GUE N , 2 N/π 2 ( dξ ) E ξ " M Y j =0 f j (Ξ( t j )) # . (4.3) 19 Since E ξ " M Y j =0 f j (Ξ( t j )) # is Φ 0 -mo derately contin uous on Y m,κ for a ny m ∈ N and κ ∈ (1 / 2 , 1) from Prop osition 2.2 (ii), (4 .3) is gua ran teed by (4.1 ) of Lemma 4.1 with M = 0 and g 0 ( ξ ) = E ξ " M Y j =0 f j (Ξ( t j )) # . Then Theorem 2.6 ( ii) is prov ed. Now w e show Theorem 2.6 (iii), whic h is deriv ed fr o m the relations: E sin h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) f 2 (Ξ( t 2 )) · · · f M (Ξ( t M ) i = E sin h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) E Ξ( t 1 ) h f 2 (Ξ( t 2 − t 1 )) · · · E Ξ( t M − 1 − t M − 2 ) h f M (Ξ( t M − t M − 1 )) i · · · ii = h f 0 , T t 1 ( f 1 T t 2 − t 1 ( f 2 · · · T t M − t M − 1 f M ) · · · ) i µ sin . (4.4 ) W e sho w the first equality of (4.4 ) b y induction with resp ect to M . F irst w e consider the case M = 2. By the Mark o v prop ert y of ( { Ξ( t ) } t ∈ [0 , ∞ ) , P µ GUE N, 2 N /π 2 ), w e ha v e E µ GUE N, 2 N /π 2 h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) f 2 (Ξ( t 2 )) i = E µ GUE N, 2 N /π 2 h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) E Ξ( t 1 ) h f 2 (Ξ( t 2 − t 1 )) ii for 0 ≤ t 1 < t 2 < ∞ . Since lim N →∞ E µ GUE N, 2 N /π 2 h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) f 2 (Ξ( t 2 )) i = E sin h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) f 2 (Ξ( t 2 )) i b y (3.12), it is enough to sho w lim N →∞ E µ GUE N, 2 N /π 2 h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) E Ξ( t 1 ) h f 2 (Ξ( t 2 − t 1 )) ii = E sin h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) E Ξ( t 1 ) h f 2 (Ξ( t 2 − t 1 )) ii (4.5) for the pro of. Since E ξ h f 2 (Ξ( t 2 − t 1 )) i = T t 2 − t 1 f 2 ( ξ ) is Φ 0 -mo derately contin uous on Y m,κ for any m ∈ N and κ ∈ (1 / 2 , 1), ( 4 .1) of Lemma 4.1 with M = 1 and g 0 = f 0 , g 1 = f 1 T t 2 − t 1 f 2 giv es (4.5 ). Th us, we ha ve obtained the case M = 2. Next w e suppose tha t the first equalit y of (4.4) is satisfied in case M = k ∈ N . By using the same argument as in the case M = 2, in whic h (4.1) of Lemma 4.1 is used in this case with M = 1 and g 0 = Q k − 1 j =0 f j , g 1 = f k T t k +1 − t k f k +1 , w e ha ve E sin " k +1 Y j =0 f j (Ξ( t j )) # = E sin " k Y j =0 f j (Ξ( t j )) E Ξ( t k ) h f k +1 (Ξ( t k +1 − t k )) i # 20 with t 0 = 0. F ro m t he assumption of the induction the right hand side of the ab o v e equalit y equals to E sin h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) E Ξ( t 1 ) h f 2 (Ξ( t 2 − t 1 )) · · · E Ξ( t k − t k − 1 ) h f k +1 (Ξ( t k +1 − t k )) i · · · ii . Then w e obtain the first equality o f (4.4 ) in case M = k + 1, and the induction is completed. The second equality of (4.4) is deriv ed from the definition (2 .14) of the op erator T t and (2.15). That is, E sin h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) E Ξ( t 1 ) h f 2 (Ξ( t 2 − t 1 )) · · · E Ξ( t M − 1 − t M − 2 ) h f M (Ξ( t M − t M − 1 )) i · · · ii = E sin h f 0 (Ξ(0)) f 1 (Ξ( t 1 )) T t 2 − t 1 ( f 2 · · · T t M − t M − 1 f M ) · · · )(Ξ( t 1 )) i = h f 0 , T t 1 ( f 1 T t 2 − t 1 ( f 2 · · · T t M − t M − 1 f M ) · · · ) i µ sin . The pro o f of Theorem 2.6 is completed. The pro of of Theorem 2.7 is obtained b y the same arg umen t, in whic h (3.14) a nd (4.2) should b e used instead of (3 .12) and (4.1). 4.3 Pro of of Lemma 4.1 W e in tro duce subsets of Y , Y γ ,L 0 κ,m =  ξ ∈ M : m ( ξ , κ ) ≤ m,     Z | x |≥ L ξ ( dx ) x     ≤ L − γ , L ≥ L 0  , with κ ∈ (1 / 2 , 1) γ > 0, m, L 0 ∈ N . Then w e can pro v e the following lemma. Lemma 4.2 (i) F or any t > 0 , lim m →∞ lim L 0 →∞ min N ∈ N µ GUE N , 2 N/π 2 + t  Y γ ,L 0 κ,m  = 1 for some κ ∈ (1 / 2 , 1 ) and γ > 0 . (ii) F or any t > 0 , lim m →∞ min N ∈ N µ ( ν ) N ,N + t  Y + κ,m  = 1 for some κ ∈ (1 , 2) . Pr o of. H ere we g iv e the pro of of (i) only , since ( ii) will b e pro v ed b y the similar argumen t to the latter ha lf of the pro o f of (i). F or an y fixed t > 0 w e put ρ N GUE ( t, x ) ≡ K N ( t, x ; t, x ) = 1 √ 2 t N − 1 X k =0 ϕ k  x √ 2 t  2 , (4.6) and ρ N ( t, x ) ≡ ρ N GUE  2 N π 2 + t, x  , 21 Then ρ N ( t, x ) is a symmetric function of x and b ounded with resp ect to N and x . Since µ GUE N , 2 N/π 2 + t is a determinan tal p oin t pro cess, by Lemma 3.1 w e ha ve Z M µ GUE N , 2 N/π 2 + t ( dξ )     ξ ([0 , L )) − Z L 0 ρ N ( t ; x ) dx     4 ≤ C L 2 with a p ositiv e constan t C , whic h is indep enden t of N . By Cheb yshev’s inequality w e ha v e µ GUE N , 2 N/π 2 + t      ξ ([0 , L )) − Z L 0 ρ N ( t ; x ) dx     ≥ L 7 / 8  ≤ C L − 3 / 2 , (4.7) and so µ GUE N , 2 N/π 2 + t      ξ ([0 , L )) − Z L 0 ρ N ( t ; x ) dx     ≤ L 7 / 8 , ∀ L ≥ L 0  ≥ 1 − C ′ L − 1 / 2 0 . Similarly , we hav e µ GUE N , 2 N/π 2 + t      ξ (( − L, 0]) − Z 0 − L ρ N ( t ; x ) dx     ≤ L 7 / 8 , ∀ L ≥ L 0  ≥ 1 − C ′ L − 1 / 2 0 . Note that Z L ≤| x |≤ L ′ ρ N ( t ; x ) dx x = 0 , L ′ > L, from the fa ct that ρ N ( t, x ) is symmetric in x . Then, by the same pro cedure in the pro of of Lemma 3.2 with C 1 = 1 and ε = 7 / 8, we ha ve µ GUE N , 2 N/π 2 + t      Z | x |≥ L ξ ( dx ) x     ≤ 24 L − 1 / 8 , ∀ L ≥ L 0  ≥ 1 − 2 C ′ L − 1 / 2 0 . (4.8) By Cheb yshev’s inequality with Lemma 3.1, fo r an y p ∈ N µ GUE N , 2 N/π 2 + t  ξ  [ g κ ( k ) , g κ ( k + 1)]  ≥ m  ≤  3 ρ N ( t, [ g κ ( k ) , g κ ( k + 1)])  p | m − ρ N ( t, [ g κ ( k ) , g κ ( k + 1)]) | 2 p ≤ C k ( κ − 1) p m − 2 p , (4.9) where ρ N ( t, D ) ≡ R D dxρ N ( t, x ) a nd C is a constant indep enden t of N and k . If p is large enough to satisfy (1 − κ ) p > 1, w e hav e µ GUE N , 2 N/π 2 + t  max k ∈ Z ξ  [ g κ ( k ) , g κ ( k + 1 )]  ≥ m  ≤ C ′ m − 2 p , and so we see lim m →∞ min N ∈ N µ GUE N , 2 N/π 2 + t  max k ∈ Z ξ  [ g κ ( k ) , g κ ( k + 1)]  ≤ m  = 1 . (4.10) 22 Com bining the ab o ve estimates (4.8) and (4.10 ) , we obtain (i) of the lemma. Pr o of of L emma 4.1. Since the pro o f s of (i) and (ii) are similar, here we only give the pro of of (i). Let d ( · , · ) b e a metric on M asso ciated with t he v ague top olo gy . First remind that ξ n con ve rges Φ 0 -mo derately to ξ if the conditions (2.3) and (2.4) are satisfied. Then f rom t he definition of Y γ ,L 0 κ,m , w e see that for an y ε > 0 there exists δ > 0 such that | g j ( ξ ) − g j ( η ) | < ε, 0 ≤ j ≤ M , for any ξ , η ∈ Y γ ,L 0 κ,m with d ( ξ , η ) < δ . Here we use the f a ct that the closure of { ξ ∈ M : m ( ξ , κ ) ≤ m } is a compact subset of M to ensure that δ do es not dep end on ξ or η . Then, from the fact (3 .12), w e can sho w that      lim N →∞ E µ GUE N, 2 N /π 2 " M Y j =0 g j (Ξ( t j )) # − E sin " M Y j =0 g j (Ξ( t j )) #      ≤ M Y j =0 sup ξ | g j ( ξ ) | ( ( M + 1) µ sin ( M \ Y γ ,L 0 κ,m ) + M X j =0 max N ∈ N µ GUE N , 2 N/π 2 + t j ( M \ Y γ ,L 0 κ,m ) ) b y using the Sk oroho d represen tatio n theorem, whic h can b e applied to distributions on P olish spaces [1]. Hence, by L emma 4.2 w e obtain the lemma. 4.4 Pro of of T h eorem 2.8 W e put ρ N A ( x ) = ρ N GUE ( N 1 / 3 , 2 N 2 / 3 + x ) , (4.11) where ρ N GUE ( t, x ) is defined in (4.6). The soft- edge scaling limit (3.1 5) implies that lim N →∞ ρ N A ( x ) = ρ Ai ( x ) ≡ K Ai ( x, x ) . W e also see the following estimates, whose pro of will b e g iven in Section 5. Lemma 4.3 Ther e exists a p ositive c onstant C 3 such that | ρ Ai ( x ) − b ρ ( x ) | ≤ C 3 | x | , x ∈ R , (4.12) and | ρ N A ( x ) − b ρ N sc ( x ) | ≤ C 3 | x | , x ∈ R , N ∈ N . (4.13) In p articular max N ∈ N Z | x |≥ L dx     ρ N A ( x ) − b ρ N sc ( x ) x     ≤ 2 C 3 L . (4.14) 23 F or the pro of of (i) w e first sho w µ Ai ( Y A ) = 1. Let ρ Ai b e the densit y function of µ Ai . Note that ρ Ai ( x ) = O ( | x | 1 / 2 ), x → −∞ from (4.12). Using Lemma 3.1 with k = 3, we see that (3.6 ) and (3.7) are satisfied with m ′ = 6 , p = 4 . 5 . Note the correlation inequalit y ρ m ( x m ) ≤ Q m j =1 ρ ( x j ) (see Prop osition 4 .3 in [32]). If w e tak e κ ∈ (1 / 2 , 2 / 3) and m ∈ N satisfying (3 κ/ 2 − 1) m < − 1, we see that the condition (3.8) is satisfied fo r the correlation functions ρ m of µ Ai . The condition (3.4) with ρ = b ρ is deriv ed from (4.12) immediately . Hence, from Lemma 3.3 (iii) w e obtain the desired result. F or the pro of of (ii) and (iii) w e introduce subsets of Y A , Y A ,γ ,L 0 κ,m = ( ξ ∈ M : m ( ξ , κ ) ≤ m,      Z | x |≥ L b ρ ξ ( R ) sc ( x ) dx − ξ ( d x ) x      ≤ L − γ , L ≥ L 0 ) with κ ∈ (1 / 2 , 2 / 3] γ > 0, m, L 0 ∈ N . Then the following lemma is established. Lemma 4.4 We h ave lim m →∞ lim L 0 →∞ min N ∈ N µ GUE N ,N 1 / 3  τ 2 N 2 / 3 Y A ,γ ,L 0 κ,m  = 1 for some κ ∈ (1 / 2 , 2 / 3) and γ > 0 . Pr o of. Note that R ∞ 0 ρ N A ( x ) dx < ∞ and R 0 − L ρ N A ( x ) dx ≤ C L 3 / 2 from Lemma 4.3. Since µ GUE N ,N 1 / 3 is a determinan tal p oint pro cess, b y Lemma 3.1 Z M τ − 2 N 2 / 3 µ GUE N ,N 1 / 3 ( dξ )     ξ ( − L, ∞ )) − Z ∞ − L ρ N A ( x ) dx     6 ≤ C L 9 / 2 with a p ositiv e constan t C , whic h is indep enden t of N . By Cheb yshev’s inequality w e ha v e τ − 2 N 2 / 3 µ GUE N ,N 1 / 3      ξ (( − L, ∞ )) − Z ∞ − L ρ N A ( x ) dx     ≥ L 23 / 24  ≤ C L 5 / 4 , (4.15) and so τ − 2 N 2 / 3 µ GUE N ,N 1 / 3      ξ (( − L, ∞ )) − Z ∞ − L ρ N A ( x ) dx     ≤ L 23 / 24 , ∀ L ≥ L 0  ≥ 1 − C ′ L 1 / 4 0 . By using the estimate (4.13) and Lemma 3.2 (ii) with C 1 = 1 and ε = 23 / 24 , w e see that τ − 2 N 2 / 3 µ GUE N ,N 1 / 3      Z | x |≥ L b ρ N sc ( x ) dx − ξ ( d x ) x     ≤ C ” L 1 / 24 , ∀ L ≥ L 0  ≥ 1 − 2 C ′ L 1 / 4 0 (4.16) with C ′′ = 72 + C 3 . Noting the estimate (4.13), by the same argumen t deriving (4.9) w e will obtain τ − 2 N 2 / 3 µ GUE N ,N 1 / 3  ξ  [ g κ ( k ) , g κ ( k + 1 )]  ≥ m  ≤ C k (3 κ/ 2 − 1) p m − 2 p . 24 and lim m →∞ min N ∈ N τ − 2 N 2 / 3 µ GUE N ,N 1 / 3  max k ∈ Z ξ  [ g κ ( k ) , g κ ( k + 1)]  ≤ m  = 1 . (4.17) Com bining the ab o ve estimates (4.16) and (4.1 7 ), w e obtain the lemma. Pr o of of (ii) and (ii i ) of The or em 2.8. F o r a n y fixed t > 0 w e put b ρ N sc ( t, x ) = s N 1 / 3 N 1 / 3 + t b ρ N sc   s N 1 / 3 N 1 / 3 + t x   , ρ N A ( t, x ) = ρ N GUE  N 1 / 3 + t, 2 N 1 / 2 ( N 1 / 3 + t ) 1 / 2 + x  , N ∈ N , and Y A ,γ ,L 0 κ,m ( t ) = ( ξ ∈ M : m ( ξ , κ ) ≤ m,      Z | x |≥ L b ρ ξ ( R ) sc ( t, x ) dx − ξ ( dx ) x      ≤ L − γ , L ≥ L 0 ) . Note that b ρ N sc ( t, · ) is a nonnegativ e function suc h that R R dx b ρ N sc ( t, x ) = N , b ρ N sc ( t, x ) ր b ρ ( x ), N → ∞ . By using the scaling prop erty of the Dyson mo del, from Lemma 4.4 we ha ve lim m →∞ lim L 0 →∞ min N ∈ N µ GUE N ,N 1 / 3 + t  τ 2 N 1 / 2 ( N 1 / 3 + t ) 1 / 2 Y A ,γ ,L 0 κ,m ( t )  = 1 , for some κ ∈ (1 / 2 , 2 / 3) and γ > 0. Note tha t 2 N 1 / 2 ( N 1 / 3 + t ) 1 / 2 − 2 N 2 / 3 + N 1 / 3 t − t 2 / 4 = O ( N − 1 / 3 ). Since     Z | x |≥ L b ρ N sc ( t, x ) dx − b ρ N sc ( t, x + ε ) dx x     ≤ εL − 1 / 2 , w e ha v e lim m →∞ lim L 0 →∞ min N ∈ N µ GUE N ,N 1 / 3 + t  τ 2 N 2 / 3 − tN 1 / 3 + t 2 / 4 Y A ,γ ,L 0 κ,m ( t )  = 1 , and so lim m →∞ lim L 0 →∞ min N ∈ N P τ − 2 N 2 / 3 µ GUE N,N 1 / 3  Ξ b ρ N sc ( t ) ∈ Y A ,γ ,L 0 κ,m ( t )  = 1 (4.18) for some κ ∈ (1 / 2 , 2 / 3) and γ > 0 . Let 0 ≡ t 0 < t 1 < t 2 < · · · < t M < ∞ , M ∈ N 0 . Supp ose that g j , 0 ≤ j ≤ M a re b ounded functions on M suc h that g j ( ξ N ) → g j ( ξ ), N → ∞ , if ξ ∈ Y A and ξ N ∈ M with ξ N ( R ) = N satisfy t hat ξ N → ξ v aguely , (2.9 ) holds with b ρ N ( · ) = b ρ N sc ( t j , · ) , and max N ∈ N m ( ξ N , κ ) ≤ m for some m ∈ N and κ ∈ (1 / 2 , 1 ). F ro m (4 .1 8), w e can sho w lim N →∞ E τ − 2 N 2 / 3 µ GUE N,N 1 / 3 " M Y j =0 g j (Ξ( t j )) # = E A i " M Y j =0 g j (Ξ( t j )) # . b y t he same a rgumen t as the pro o f of Lemma 4.1. Therefore, b y applying (ii) o f Prop o sition 2.5 w e can sho w (ii) and (iii) of Theorem 2.8 b y the same pro cedure as in the pr o of of Theorem 2.6. 25 5 Pro of of Lemma 4. 3 5.1 Pro of of (4.12) W e use the fo llo wing asymptotic expansions of the Airy functions in t he classical sense of P oincar´ e [37, 2 5 , 26]: F o r x ≫ 1 Ai( x ) ≈ e − 2 3 x 3 / 2 2 π 1 / 2 x 1 / 4 L  − 2 3 x 3 / 2  , Ai ′ ( x ) ≈ − x 1 / 4 e − 2 3 x 3 / 2 2 π 1 / 2 M  − 2 3 x 3 / 2  , Ai( − x ) ≈ 1 π 1 / 2 x 1 / 4  sin  2 3 x 3 / 2 − π 4  Q  2 3 x 3 / 2  + cos  2 3 x 3 / 2 − π 4  P  2 3 x 3 / 2  , Ai ′ ( − x ) ≈ x 1 / 4 π 1 / 2  sin  2 3 x 3 / 2 − π 4  R  2 3 x 3 / 2  − cos  2 3 x 3 / 2 − π 4  S  2 3 x 3 / 2  , where L ( z ) , M ( z ) , P ( z ) , Q ( z ) , R ( z ) and S ( z ) a re functions defined by L ( z ) = ∞ X k =0 u k z k = 1 + 5 72 z + O  1 z 2  , M ( z ) = ∞ X k =0 v k z k = 1 − 7 72 z + O  1 z 2  , P ( z ) = ∞ X k =0 ( − 1) k u 2 k z 2 k = 1 + O  1 z 2  , Q ( z ) = ∞ X k =0 ( − 1) k u 2 k + 1 z 2 k + 1 = 5 72 z + O  1 z 3  , R ( z ) = ∞ X k =0 ( − 1) k v 2 k z 2 k = 1 + O  1 z 2  , S ( z ) = ∞ X k =0 ( − 1) k v 2 k + 1 z 2 k + 1 = − 7 72 z + O  1 z 3  , with u k = Γ(3 k + 1 / 2 ) 54 k k !Γ( k + 1 / 2) , v k = − 6 k + 1 6 k − 1 u k . Then ρ Ai ( x ) = ( Ai ′ ( x )) 2 − x (Ai( x )) 2 = O  e − 4 3 x 3 / 2  , x → ∞ , and ρ Ai ( − x ) = (Ai ′ ( − x )) 2 + x (Ai( − x )) 2 = √ x π "  sin  2 3 x 3 / 2 − π 4   1 + O  1 x 3  − cos  2 3 x 3 / 2 − π 4   − 7 72 x 3 / 2 + O  1 x 9 / 2  2 +  sin  2 3 x 3 / 2 − π 4   5 72 x 3 / 2 + O  1 x 9 / 2  + cos  2 3 x 3 / 2 − π 4   1 + O  1 x 3  2 # = √ x π " 1 + sin  2 3 x 3 / 2 − π 4  cos  2 3 x 3 / 2 − π 4  1 3 x 3 / 2 + O  1 x 3  # , x → ∞ . Hence, w e obtain     ρ Ai ( x ) − b ρ ( x )     = O  1 | x |  , x → ∞ . 26 5.2 Pro of of (4.13) F or provin g (4 .1 3) w e use the asymptotic b eha vior of Hermite p olynomials giv en in Planc herel and Rotach [29], in whic h the Hermite p olynomials are defined a s b H n ( x ) = ( − 1) n e x 2 / 2 ( d n /dx n ) e − x 2 / 2 , whereas our definition is H n ( x ) = ( − 1) n e x 2 ( d n /dx n ) e − x 2 . W e should note the relation that b H n ( x ) = 2 − n/ 2 H n  x/ √ 2  . W e in tro duce the following po lynomials φ n ( z ) and ψ np ( z ) deter- mined by the expansions exp h z ∞ X m =3 ( − 1) m m τ m − 2 i = ∞ X n =0 φ n ( z ) τ n , ∞ X k =1 1 k τ k − 1 ! p exp h z ∞ X m =4 1 m τ m − 3 i = ∞ X n =0 ψ np ( z ) τ n , for | τ | < 1 . F or example, φ 0 ( z ) = 1, φ 1 ( z ) = − z / 3, φ 2 ( z ) = z / 4 + z 2 / 18, φ 3 ( z ) = − z / 5 − z 2 / 12 − z 3 / 162, ψ 0 p ( z ) = 1, ψ 1 p ( z ) = p/ 2 + z / 4. W e set the co efficien ts of these po lynomials as φ n ( z ) = P n m =0 a nm z m and ψ np ( z ) = P n m =0 b ( p ) nm z m . F or example, a 00 = 1 , a 10 = 0 , a 11 = − 1 3 , a 20 = 0 , a 21 = 1 4 , a 22 = 1 18 , a 30 = 0 , a 31 = − 1 5 , a 32 = − 1 12 , a 33 = − 1 162 , b ( p ) 00 = 1 , b ( p ) 10 = p 2 , b ( p ) 11 = 1 4 . Then the a symptotic b eha viors of Hermite p olynomials g iv en in [29] are summarized as follo ws. (A.1) When x 2 < 2( N + 1), we put x = p 2( N + 1) cos θ , (0 < θ ≤ π / 2). Then f or a n y L ∈ N H N ( x ) N ! = 2 N/ 2 exp  ( N + 1)(1 / 2 + cos 2 θ )  ( N + 1) ( N + 1) / 2 ( π sin θ ) 1 / 2 × " L − 1 X n =0 n X m =0 C 1 nm ( N , θ ) sin  N + 1 2 (2 θ − sin 2 θ ) + D 1 nm ( θ )  + O  ( N sin 3 θ ) − L/ 2  # , where C 1 nm ( N , θ ) = 1 + ( − 1) n 2 Γ( n + n +1 2 ) ( N + 1) n/ 2 (sin θ ) m + n/ 2 a nm , and D 1 nm ( θ ) = π 4 − θ 2 − (2 m + n )  π 4 + θ 2  . (A.2) When x 2 > 2( N + 1 ) , w e put x = p 2( N + 1) cosh θ , (0 < θ < ∞ ) . Then for an y L ∈ N H N ( x ) N ! = 2 N/ 2 exp  ( N + 1)(1 / 2 + cosh θ (cosh θ − sinh θ )  ( N + 1) ( N + 1) / 2 (2 π sinh θ ) 1 / 2 (cosh θ − sinh θ ) ( N + 1 / 2) × " L − 1 X n =0 n X m =0 C 2 nm ( θ , N ) + O  N − L/ 2 θ − 3 L/ 2 )  # , 27 where C 2 nm ( θ , N ) = 1 + ( − 1) n 2 Γ( n + n +1 2 ) ( N + 1) n/ 2  − 2 1 − e − 2 θ  m + n/ 2 a nm . (A.3) When x 2 ∼ 2 ( N + 1), w e put x = p 2( N + 1) − 2 − 1 / 2 N − 1 / 6 y , y = o ( N 2 / 3 ). Then there exists a p ositiv e constan t h ∗ suc h H N ( x ) N ! = e 3 x 2 / 4 π  x/ √ 2  N +2 / 3 " ∞ X p =0 A p ( x ) p ! y p + O  x − 1 / 3 e − h ∗ x 2  # with the function A p ha ving the following asymptotic expansions: A p ( x ) = 3 ( p − 2) / 3 L − 1 X n =0 n X m =0 ( − 1) m 3 m + n/ 3  x/ √ 2  2 n/ 3 Γ  p + n + 1 3 + m  sin  p + n + 1 3 π  b ( p ) nm + O  x − 2 L/ 3  . Applying the a b o v e to the function ϕ N ( x ) defined in (3.10 ), w e obtain the lemma. Lemma 5.1 (i) F or N ∈ N , θ ∈ [0 , π / 2) and L ∈ N ϕ N  p 2( N + 1) cos θ  = 1 + O ( N − 1 ) √ π sin θ  2 N  1 / 4 × " L − 1 X n =0 n X m =0 C 1 nm ( N , θ ) sin  N + 1 2 (2 θ − sin 2 θ ) + D 1 nm ( θ )  + O  ( N sin 3 θ ) − L/ 2  # . (5.1) (ii) F or N ∈ N an d θ ∈ [0 , ∞ ) ϕ N  p 2( N + 1) cosh θ  = 1 + O ( N − 1 ) √ 2 π sinh θ  1 2 N  1 / 4 exp  N + 1 2  (2 θ − sinh 2 θ )  × " L − 1 X n =0 n X m =0 C 2 nm ( θ , N ) + O  N − L/ 2 θ − 3 L/ 2 )  # . (5.2) (iii) F or N ∈ N an d y w i th | y | = o ( N 2 / 3 ) ϕ N  p 2( N + 1) − y √ 2 N 1 / 6  = 2 1 / 4 N − 1 / 12  B  y , p 2( N + 1) − y √ 2 N 1 / 6  + O ( N − 1 )  , (5.3) wher e B ( y , x ) ≡ Ai( − y ) +  x 2  − 2 / 3 n c 10 Ai ′ ( − y ) + c 11 y 2 Ai( − y ) o +  x 2  − 4 / 3 n c 20 Ai ′ ( − y ) + c 21 y 2 Ai( − y ) + c 22 y 3 Ai ′ ( − y ) o with c onstants c nm , 0 ≤ m ≤ n ≤ 2 , w h ich do not dep end on x and y . 28 Pr o of. Applying Stirling’s fo r mula N ! =  N e  N √ 2 π N  1 + 1 12 N + O ( N − 2 )  to the results (A.1) and (A.2) , w e obtain ( 5.1) and (5.2) by simple calculatio n. T o obtain (5.3) w e use (A.3) with L = 3 and the expansion Ai( − y ) = 1 π ∞ X p =0 3 ( p − 2) / 3 Γ  p + 1 3  sin  2( p + 1) 3 π  ( − y ) p p ! = 1 π ∞ X p =0 3 ( p − 2) / 3 Γ  p + 1 3  sin  p + 1 3 π  y p p ! giv en in (2.3 6 ) in [37]. W e sho w the main estimate in this section. Lemma 5.2 (i) L et x = √ 2 N cos θ with N ∈ N a nd θ ∈ (0 , π / 2) . Supp ose that N sin 3 θ ≥ C N ε for some C > 0 and ε > 0 . The n N − 1 X k =0  ϕ k ( x )  2 = ρ N sc (1 , x ) + O  1 √ N sin 2 θ  . (ii) L et x = √ 2 N cosh θ with N ∈ N and θ > 0 . Supp ose that N sinh 3 θ ≥ N ε for some C > 0 and ε > 0 . Then N − 1 X k =0  ϕ k ( x )  2 = O  1 √ N sinh 2 θ  . (iii) L et x = 2 N 2 / 3 − y with N ∈ N and | y | ≤ C N β for some C > 0 and β ∈ (0 , 2 / 21) ρ N GUE ( N 1 / 3 , x ) = ρ Ai ( − y ) + O  | y | − 1  . Pr o of. F or t he pro of of t his lemma w e use the Christoffel-Da rb oux formula N − 1 X k =0  ϕ k ( x )  2 = N  ϕ N ( x )  2 − p N ( N + 1) ϕ N +1 ( x ) ϕ N − 1 ( x ) = N n ϕ N ( x )  2 − ϕ N +1 ( x ) ϕ N − 1 ( x ) o  1 + O ( N − 1 )  . (5.4) F or prov ing (i) we first sho w that for ℓ ∈ {− 1 , 0 , 1 } ϕ N + ℓ  √ 2 N cos θ  = 1 + O ( N − 1 ) √ π sin θ  2 N  1 / 4 × " L − 1 X n =0 n X m =0 C 1 nm ( N − 1 , θ ) sin  N 2 (2 θ − sin 2 θ ) + D 1 nm ( θ ) − (1 + ℓ ) θ  + O  1 N sin θ  # . (5.5) 29 Substituting N − 1 instead of N in (5.1) and taking L > 2 /ε , we hav e (5.5) with ℓ = − 1. F or calculating ϕ N + ℓ ( x ) with ℓ ∈ { 0 , 1 } , w e tak e η ℓ suc h that cos( θ + η ℓ ) = s 2 N 2( N + 1 + ℓ ) cos θ , and then ϕ N + ℓ  √ 2 N cos θ  = ϕ N + ℓ  p 2( N + 1 + ℓ ) cos( θ + η ℓ )  . (5.6) Since s 2 N 2( N + 1 + ℓ ) = r 1 − 1 N + 1 + ℓ = 1 − 1 + ℓ 2 N + O  1 N 2  and cos( θ + η ℓ ) = cos θ − η ℓ sin θ + O ( η 2 ℓ ) , w e ha v e η ℓ = (1 + ℓ ) cos θ 2 N sin θ + O  1 N 2 sin θ  = O  1 N sin θ  . (5.7) F rom (5.1) and ( 5 .6), w e obtain ϕ N + ℓ  √ 2 N cos θ  = 1 + O ( N − 1 ) √ π sin θ  2 N  1 / 4 × " L − 1 X n =0 n X m =0 C 1 nm ( N + ℓ, θ + η ℓ ) sin  N 2  2( θ + η ℓ ) − sin 2( θ + η ℓ )  + D 1 nm ( θ + η ℓ )  + O  ( N sin 3 θ ) − L/ 2  # . (5.8) By simple calculations with (5.7), we ha ve N 2 n sin 2( θ + η ℓ ) − 2( θ + η ℓ ) o = N 2 (sin 2 θ − 2 θ ) − (1 + ℓ ) θ + O  1 N sin θ  . Then from (5.8) with L > 2 /ε and the ab o ve estimate, w e obtain (5.5 ) for ℓ ∈ { 0 , 1 } . Substituting (5.5 ) to (5.4)C fro m the iden tity 2 sin A sin B − sin( A + θ ) sin ( B − θ ) − sin ( A − θ ) sin( B + θ ) = 2 sin 2 θ cos( A − B ) with A = N (2 θ − sin 2 θ ) / 2 + D 1 nm ( θ ) − θ and B = N (2 θ − sin 2 θ ) / 2 + D 1 n ′ m ′ ( θ ) − θ , 0 ≤ m ≤ n ≤ L − 1, 0 ≤ m ′ ≤ n ′ ≤ L − 1, w e hav e N − 1 X k =0  ϕ k ( √ 2 N cos θ )  2 = √ 2 N π sin θ + O  1 √ N sin 2 θ  . 30 Since sin θ = √ 1 − cos 2 θ = p 1 − x 2 / { 2( N + 1) } = p 2( N + 1) − x 2 / p 2( N + 1), N − 1 X k =0  ϕ k ( x )  2 = 1 π √ 2 N − x 2 + O  1 √ N sin 2 θ  = ρ N sc (1 , x ) + O  1 √ N sin 2 θ  . (5.9) Hence, the pro of of (i) is complete. F or prov ing (ii) we sho w tha t for ℓ ∈ {− 1 , 0 , 1 } ϕ N + ℓ  √ 2 N cosh θ  = 1 + O ( N − 1 ) √ 2 π sinh θ  1 2 N  1 / 4 × exp  N + 1 + ℓ 2  (2 θ − sinh 2 θ ) + (1 + ℓ ) θ  × " L − 1 X n =0 n X m =0 C 2 nm ( θ , N + ℓ ) + O  cosh 3 θ N sinh θ  # . (5.10) Substituting N − 1 instead of N in (5.2 ) a nd taking L > 2 /ε , w e hav e (5 .1 0) with ℓ = − 1. F or ℓ ∈ { 0 , 1 } , tak e η ℓ suc h that cosh( θ + η ℓ ) = s 2 N 2( N + 1 + ℓ ) cosh θ , and w e ha ve ϕ N +1  √ 2 N cosh θ  = ϕ N +1  p 2( N + 1 + ℓ ) cosh( θ + η ℓ )  . Since cosh( θ + η ℓ ) = cosh θ + η ℓ sinh θ + O ( η 2 ℓ ), w e ha v e η ℓ = − (1 + ℓ ) cosh θ 2 N sinh θ + O  1 N 2 sinh θ  = O  cosh θ N sinh θ  . Substituting the equalit y to (5.2)Cw e hav e ϕ N + ℓ  √ 2 N cosh θ  = 1 + O ( N − 1 ) √ 2 π sinh θ  1 2 N  1 / 4 × exp  N + 1 + ℓ 2  n 2( θ + η ℓ ) − sinh 2( θ + η ℓ ) o  × " L − 1 X n =0 n X m =0 C 2 nm ( θ + η ℓ , N + ℓ ) + O  N − L/ 2 θ − 3 L/ 2 )  # . Hence if w e take L ∈ N suc h that L > 2 /ε , from the r elat io n 2( θ + η ℓ ) − sinh 2( θ + η ℓ ) = 2 θ − sinh 2 θ + 1 + ℓ N sinh 2 θ + O  cosh 3 θ N 2 sinh θ  , 31 w e can conclude (5 .10) with ℓ ∈ { 0 , 1 } . Com bining (5.10) with (5.4), w e obtain N − 1 X k =0  ϕ k  √ 2 N cosh θ  2 = N 2 π sinh θ 1 √ 2 N exp h ( N + 1)(2 θ − sinh 2 θ ) + 2 θ i × O  cosh 3 θ N sinh θ  . Then (ii) is concluded b y simple calculation. Finally , we sho w (iii). Put w = √ 2 N − 2 − 1 / 2 N − 1 / 6 y , F ro m the Christoffel-D arb oux form ula (5.4) and (5.3) ϕ N − 1  √ 2 N − y √ 2 N 1 / 6  = 2 1 / 4 N − 1 / 12 n B  y (1 − N − 1 ) 1 / 6 , √ 2 N − y √ 2 N 1 / 6  + O ( N − 1 ) o = 2 1 / 4 N − 1 / 12 n B ( y , w ) + O ( N − 1 ) o . Noting the fo llo wing tw o simple relations, √ 2 N − cN − 1 / 6 = p 2( N + 1 + ℓ ) −  c + 1 + ℓ √ 2 N 1 / 3  N − 1 / 6 + O ( N − 3 / 2 ) , ℓ ∈ { 0 , 1 } , and y / √ 2 + (1 + ℓ ) / { √ 2 N 1 / 3 } = { y + (1 + ℓ ) N − 1 / 3 } / √ 2, w e apply (5.3) to obtain ϕ N + ℓ  √ 2 N − y √ 2 N 1 / 6  = 2 1 / 4 N − 1 / 12  B  y + (1 + ℓ ) N − 1 / 3 , w − ℓ + 1 √ 2 N  + O ( N − 1 )  . Hence ϕ N ( w ) 2 − ϕ N +1 ( w ) ϕ N − 1 ( w ) = √ 2 N − 1 / 6 " B  y + N − 1 / 3 , w − 1 √ 2 N  2 − B  y + 2 N − 1 / 3 , w − 2 1 √ 2 N  B ( y , w ) # + O ( N − 7 / 6 ) . Noting that B  y + N − 1 / 3 , w − 1 √ 2 N  2 − B  y + 2 N − 1 / 3 , w − 2 1 √ 2 N  B ( y , w ) = (  ∂ ∂ y B ( y , w )  2 − B ( y , w ) ∂ 2 ∂ y 2 B ( y , w ) ) N − 2 / 3  1 + O  y 1 / 2 w N 1 / 6  32 and the asymptotic prop erties of B ( y , x ) − Ai( − y ) and its deriv ativ es given b y B ( y , x ) − Ai( − y ) = O ( x − 2 / 3 y 7 / 4 + x − 4 / 3 y 13 / 4 ) ∂ k ∂ y k ∂ ℓ ∂ x 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 ) , x, y → ∞ , w e obt a in 1 √ 2 N 1 / 6 N − 1 X k =0 ϕ k  √ 2 N − y √ 2 N 1 / 6  2 = (  ∂ ∂ y B ( y , w )  2 − B ( y , w ) ∂ 2 ∂ y 2 B ( y , w ) )  1 + O  y 1 / 2 w N 1 / 6  = (  ∂ ∂ y Ai( − y )  2 − Ai( − y ) ∂ 2 ∂ y 2 Ai( − y ) + O ( y 5 / 2 w − 2 / 3 ) )  1 + O  y 1 / 2 w N 1 / 6  =  ∂ ∂ y Ai( − y )  2 − Ai( − y ) ∂ 2 ∂ y 2 Ai( − y ) + O ( y 5 / 2 N − 1 / 3 ) . Hence w e conclude tha t f or x = √ 2 N 1 / 6 w = 2 N 2 / 3 − y , | y | ≤ N β with some β ∈ (0 , 2 / 21), ρ N GUE ( N 1 / 3 , x ) =  ∂ ∂ y Ai( − y)  2 − Ai( − y ) ∂ 2 ∂ y 2 Ai( − y ) + O  | y | − 1  . This completes the pro of. Putting x = 2 N 2 / 3 (cos θ − 1) ∈ ( −∞ , 0). w e ha ve − x ∼ N 2 / 3 θ 2 . In case N sin 3 θ ≥ N 3 ε/ 2 for some ε > 0, w e see from Lemma 5.2 (i) that ρ N A ( x ) = b ρ N sc ( x ) + O  1 N 2 / 3 θ 2  = b ρ N sc ( x ) + O  | x | − 1  . Similarly , Putting x = 2 N 2 / 3 (cosh θ − 1) ∈ (0 , ∞ ), w e ha ve x ∼ N 2 / 3 (cosh θ − 1). In case N sinh 3 θ ≥ N 3 ε/ 2 for some ε > 0, w e see from Lemma 5.2 (ii) that ρ N A ( x ) = b ρ N sc ( x ) + O  1 N 2 / 3 sinh 2 θ  = b ρ N sc ( x ) + O  | x | − 1  . In the case t hat | x | ≤ N ε for some ε ∈ (0 , 2 / 21), w e see from Lemma 5 .2 (ii) that ρ N A ( x ) = ρ Ai ( x ) + O  | x | − 1  = b ρ N sc ( x ) + | b ρ N sc ( x ) − b ρ ( x ) | + | b ρ ( x ) − ρ Ai ( x ) | + O  | x | − 1  . 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