Limit Theorems for Interacting Brownian Motions

Limi Theorems for Ineracing Brownian Moions Makoo Kaori 10 June 2018 version 2a Absrac Dyson model is a sochasic paricle sysem in one dimension R, in which repulsive force acs beween any pair of paricles wih srengh proporional o he inverse of disance. This mulivariae sochasic process is realized as he sysem of one-dimensional Brownian moions condiioned never o collide wih each oher. We can show ha his many-body sysem is exacly solvable and of deerminanal in he sense ha any spaioemporal correlaion funcion is expressed by deerminan and is conrolled by a single coninuous funcion called he correlaion kernel. In his lecure, we assume he special iniial configuraion such ha all paricles are concenraed on he origin and we discuss he limi heorems in. Wigner s semicircle law, which is exensively sudied in random marix heory and free probabiliy, is demonsraed as he law of large numbers LL, which describes he densiy profile of paricles in R a each ime. Two kinds of limis called he bulk scaling limi and he sof-edge scaling limi are inroduced in order o obain deerminanal processes wih an infinie number of paricles. As he cenral limi heorem CLT associaed wih he laer scaling limi, he Tracy Widom disribuion is discussed. Key words: Dyson model; noncolliding Brownian moions; deerminanal processes; Fredholm deerminans; Wigner s semicircle law; random marix heory; scaling limis; Tracy-Widom disribuion; Painlevé II equaion Conens 1 Inroducion 2 1.1 Brownian moion........................................... 2 1.2 Karlin-McGregor-Lindsröm-Gessel-Vienno formula....................... 6 1.3 Dyson model as noncolliding Brownian moions.......................... 7 1.4 Fredholm deerminans and deerminanal processes....................... 11 This manuscrip was prepared for Mini Workshop : Modern Theory of Sochasic Paricles in 27-28 June 2018 held a Wroc law Universiy of Technology organized by Jacek Ma lecki and Pior Graczyk. Fakulä für Mahemaik, Universiä Wien, Oskar-Morgensern-Plaz 1, A-1090 Wien, Ausria. On sabbaical leave from Deparmen of Physics, Faculy of Science and Engineering, Chuo Universiy, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail: kaori@phys.chuo-u.ac.jp 1

2 Wigner s Semicircle Law as LL 12 2.1 Hermie orhonormal funcions and BM.............................. 12 2.2 Dyson model saring from ξ = δ 0 and Hermie kernel..................... 15 2.3 Wigner s semicircle law....................................... 16 3 Scaling Limis and Infinie Paricle Sysems 22 3.1 Bulk scaling limi and sine kernel.................................. 22 3.2 Sof-edge scaling limi and Airy kernel............................... 23 4 Tracy Widom Disribuion as CLT 26 4.1 Disribuion funcion of he maximum posiion of paricles.................... 26 4.2 Inegrals involving resolven of correlaion kernel......................... 29 4.3 onlinear hird-order differenial equaion............................. 30 4.4 Sof-edge scaling limi........................................ 32 4.5 Painlevé II and limi heorem of Tracy and Widom........................ 33 1 Inroducion 1.1 Brownian moion We consider he moion of a Brownian paricle in one-dimensional space R, saring from he origin 0 a ime = 0. A each ime > 0, he paricle posiion is randomly disribued, and each realizaion of is pah rajecory is denoed by ω and called a sample pah or simply a pah. Le Ω be he collecion of all sample pahs and we call i he sample pah space. The posiion of he Brownian paricle a ime 0 in a pah ω Ω is wrien as B, ω. Usually we omi ω and simply wrie i as B, 0. We represen each even associaed wih he process by a subse of Ω, and he collecion of all evens is denoed by F. The whole sample pah space Ω and he empy se are in F. For any wo ses A, B F, we assume ha A B F and A B F. If A F, hen is complemen A c is also in F. I is closed for any infinie sum of evens in he sense ha, if A n F, n = 1, 2,..., hen n 1 A n F. Such a collecion is said o be a σ-field sigma-field. The symbol σ is for sum. A probabiliy measure P is a nonnegaive funcion defined on he σ-field F. Since any elemen of F is given by a se as above, any inpu of P is a se; P is a se funcion. I saisfies he following properies: P[A] 0 for all A F, P[Ω] = 1, P[ ] = 0, and if A, B F are disjoin, A B =, hen P[A B] = P[A] + P[B]. In paricular, P[A c ] = 1 P[A] for all A F. The riple Ω, F, P is called he probabiliy space. The smalles σ-field conaining all inervals on R is called he Borel σ-field and denoed by BR. A random variable or measurable funcion is a real-valued funcion fω on Ω such ha, for every Borel se A BR, f 1 A F. Two evens A and B are said o be independen if P[A B] = P[A]P[B]. Two random variables X and Y are independen if he evens A = {X : X A} and B = {Y : Y B} are independen for any A, B BR. The one-dimensional sandard Brownian moion, {B, ω : 0}, has he following hree properies: BM1 B0, ω = 0 wih probabiliy one. 2

BM2 There is a subse of he sample pah space Ω Ω, such ha P[ Ω] = 1 and for any ω Ω, B, ω is a real coninuous funcion of. We say ha B has a coninuous pah almos surely a.s., for shor. BM3 For an arbirary M {1, 2, 3,... }, and for any sequence of imes, 0 0 < 1 < < M, he incremens B m B m 1, m = 1, 2,..., M, are independen, and each incremen is in he normal disribuion he Gaussian disribuion wih mean 0 and variance σ 2 = m m 1. I means ha for any 1 m M and a < b, where we define for x, y R [ ] P B m B m 1 [a, b] = p, y x = b a p m m 1, z 0dz, 1 2π e x y2 /2, for > 0, δx y, for = 0. 1.1 Unless oherwise noed, he one-dimensional sandard Brownian moion is simply abbreviaed o BM in his lecure noe. The probabiliy measure P for he BM is called he Wiener measure. The expecaion wih respec o he probabiliy measure P is denoed by E. We wrie he condiional probabiliy as P[ C], where C denoes he condiion. The condiional expecaion is similarly wrien as E[ C]. The hird propery BM3 given above implies ha for any 0 s < [ ] P B A Bs = x = p s, y xdy 1.2 holds, A BR, x R. Therefore he inegral kernel p, y x given by 1.1 is called he ransiion probabiliy densiy funcion of Brownian moion saring from x. The probabiliy ha he BM is observed in a region A m BR a ime m for each m = 1, 2,..., M is hen given by [ ] P B m A m, m = 1, 2,..., M = dx 1 dx M A 1 A M A M p m m 1, x m x m 1, 1.3 where x 0 0. By BM3, we can see ha, for any c > 0, he probabiliy disribuion of Bc 2 /c is equivalen o ha of B a arbirary ime 0. I is wrien as m=1 1 c Bc2 d = B, c > 0, where he symbol d = is for equivalence in disribuion. Moreover, 1.3 implies ha, for any c > 0, B, 0 and is ime-changed process wih c 2 muliplied by a facor 1/c 3

dilaaion follow he same probabiliy law. This equivalence in probabiliy law of sochasic processes is expressed as law 1 B 0 = c Bc2, c 0, 1.4 0 and called he scaling propery of Brownian moion. For a > 0, le T a = inf{ 0 : B = a}. Then for any 0, P[T a <, B < a] = P[T a <, B > a], 1.5 since he ransiion probabiliy densiy 1.1 is a symmeric funcion of he incremen y x. This propery is called he reflecion principle of BM. For {ω : B > a} {ω : T a < }, a > 0, he above is equal o P[B > a]. The formula 1.3 also means ha for any fixed s 0, under he condiion ha Bs is given, {B : s} and {B : > s} are independen. This independence of he evens in he fuure and hose in he pas is called he Markov propery. A posiive random variable τ is called sopping ime or Markov ime, if he even {ω : τ } is deermined by he behavior of he process unil ime and independen of ha afer. For any sopping ime τ, {B : τ} and {B : > τ} are independen. I is called he srong Markov propery. A sochasic process which has he srong Markov propery and has a coninuous pah almos surely is generally called a diffusion process. For each ime [0,, we wrie he smalles σ-field generaed by he BM up o ime 0 as σbs : 0 s and define F σbs : 0 s, 0. 1.6 By definiion, wih respec o any even in F, Bs is measurable a every s [0, ]. Then we have a nondecreasing family {F : 0} of sub-σ-fields of he original σ-field F in he probabiliy space Ω, F, P such ha F s F F for 0 s < <. We call his family of σ-fields a filraion. The BM sared a x R, which is denoed by B x, 0, is defined by B x = x + B, x R, 0. 1.7 We define P x [B ] = P[B x ] and E x [fb] = E[fB x ] for any bounded measurable funcion f, 0. The sopping ime τ menioned above can be defined using he noion of filraion as follows: {ω : τ } F, 0. The srong Markov propery of BM is now expressed as E[fBs + F s ] = E Bs [fb], 0, a.s., 1.8 provided ha s 0 is any realizaion of a sopping ime τ and f is an arbirary measurable bounded funcion. 4

Since he probabiliy densiy of incremen in any ime inerval s > 0, p s, z 0, has mean zero, BM saisfies he equaliy E[B F s ] = Bs, 0 s < <, a.s. 1.9 Tha is, he mean is consan in ime, even hough he variance increases in ime as σ 2 =. Processes wih such a propery are called maringales. We noe ha for 0 s < <, E[B 2 F s ] = E[B Bs 2 + 2B BsBs + Bs 2 F s ] = E[B Bs 2 F s ] + 2E[B BsBs F s ] + E[Bs 2 F s ]. By he propery BM3 and he definiion of F s, Then we have he equaliy E[B Bs 2 F s ] = s, E[B BsBs F s ] = E[B Bs F s ] Bs = 0, E[Bs 2 F s ] = Bs 2. E[B 2 F s ] = Bs 2 s, 0 s < <, a.s. 1.10 I means ha B 2 is a maringale. For he ransiion probabiliy densiy of BM 1.1, i should be noed ha p, y x = p, x y for any x, y R, and u x p, y x is a unique soluion of he hea equaion diffusion equaion u x = 1 2 2 x u x, x R, 0 1.11 2 wih he iniial condiion u 0 x = δx y. The soluion of 1.11 wih he iniial condiion u0x f = fx, x R is hen given by u f x = E x [fb] = fyp, y xdy, 1.12 if f is a measurable funcion saisfying he condiion e ax2 fx dx < for some a > 0. Since p, y x plays as an inegral kernel in 1.12, i is also called he hea kernel. For 0 s < <, ξ R, consider E[e 1ξB Bs F s ]. Using p, i is calculaed as follows: The obained funcion of ξ R, e 1ξz p s, z 0dz = e /2 s 1ξz e z2 = e ξ2 s/2. 2π s dz E[e 1ξB Bs F s ] = e ξ2 s/2, 0 s < <, 1.13 is called he characerisic funcion of BM. 5

1.2 Karlin-McGregor-Lindsröm-Gessel-Vienno formula Le denoe he spaial dimension. For 2, he -dimensional BM in R saring from he posiion x = x 1,..., x R is defined by he -dimensional vecor-valued diffusion process, B x = B x 1 1, B x 2 2,..., B x, 0, 1.14 where {B x i i } i=1, 0 are independen one-dimensional BMs. Consider a subspace of R defined by W A {x = x 1, x 2,..., x R : x 1 < x 2 < < x }, 1.15 which is called he Weyl chamber of ype A 1 in represenaion heory. As a mulidimensional exension of he absorbing Brownian moion in R + = {x R : x > 0} wih an absorbing wall a x = 0 see, for insance, [4], we consider he absorbing Brownian moion B x = B x 1 1,..., B x in W A. The saring poin x is assumed o be in W A. We pu absorbing walls a he boundaries of WA. When Bx his any of he walls, i is absorbed and he process is sopped. In oher words, he Brownian moion B x sared a x W A is killed when i arrives a he boundaries of WA. We define q, y x for x, y W A, 0 as he probabiliy densiy for he even such ha his absorbing Brownian moion saring from x a ime = 0 survives up o ime and arrives a y a he ime. oe ha he boundaries of W A are he hyperplanes x i = x j, 1 i < j in R. Then, if we inerpre x R as a configuraion of paricles on a line R, his absorbing Brownian moion in W A can be regarded as an - paricle sysem such ha each paricle execues BM when disances beween neighboring paricles are posiive, bu when any wo neighboring paricles collide, he process is sopped. This process is a coninuum limi diffusion scaling limi [26, 27] of he vicious walker model on Z inroduced by Fisher [9] see also [6]. The following is known as he Karlin McGregor formula [22]. oe ha he discree analogue is known as he Lindsröm Gessel Vienno formula [32, 12]. Lemma 1.1 The ransiion probabiliy densiy of he absorbing Brownian moion in W A is given by [ ] q, y x = de p, y i x j 1 i,j = sgnσ p, y σi x i, x, y W A, 0, 1.16 σ S i=1 where S denoes he se of all permuaions of indices {1, 2,..., }. Proof By he propery BM2 and he definiion of he ransiion probabiliy densiy of BM, p, y x gives he oal probabiliy mass of he Brownian pah π[0, ] from x o y wih ime duraion. Le Π x, y denoe he collecion of all Brownian pahs from x R o y R wih ime duraion 0. We will inerpre 1.16 as a generaing funcion for +1-uples, 6

σ, π 1,..., π, where σ S, π i = π i [0, ] Π x i, y σi, 1 i. Under he assumpion x W A, le τ = inf{ > 0 : B x / W A }. 1.17 Assume ha τ < and B x k k τ = Bx l l τ. For a pair of pahs π k, π l, we define π k, π l by exchanging he Brownian pahs of π k, π l afer = τ: π k[0, ] = π k [0, τ] π l τ, ], π l[0, ] = π l [0, τ] π k τ, ]. We define π i = π i for i k, l and σ = σ σ kl, where σ kl denoes he exchange of k and l. Then he operaion σ, π 1,..., π σ, π 1,..., π is an involuion. By his operaion, he absolue value of he conribuion o he generaing funcion 1.16 is no changed because of he srong Markov propery 1.8 and he reflecion principle of BM 1.5, bu he sign is changed. So he conribuion of any such pairs {σ, π 1,..., π, σ, π 1,..., π } is canceled ou. The remaining non-zero conribuions in 1.16 are from -uples of noninersecing Brownian pahs. Since x, y W A, σ = id and so sgnσ = sgnid = 1 for noninersecing pahs. Hence 1.16 gives he oal probabiliy mass of -uples of noninersecing Brownian pahs from x o y wih ime duraion and is idenified wih q, y x for he absorbing Brownian moion in W A. 1.3 Dyson model as noncolliding Brownian moions For an iniial configuraion x W A, he survival probabiliy of he absorbing Brownian moion in W A up o ime 0 is hen given by P x [τ > ] = q, y xdy, 0, 1.18 W A where dy = i=1 dy i. ow we consider an -paricle sysem of BMs in R condiioned never o collide wih each oher, ha is, hey do no collide even during he ime inerval,. We simply call his process he noncolliding Brownian moion wih paricles. The ransiion probabiliy densiy funcion p, y x of his process from x W A o y WA wih ime duraion 0 should be obained by he following limi, p, y x = lim T q, y xp y [τ > T ] P x. 1.19 [τ > T ] Le Then he following are proved. h x = x j x i. 1.20 1 i<j 7

Proposiion 1.2 i The ransiion probabiliy densiy of he noncolliding Brownian moion wih paricles is given by p, y x = h y h x q, y x, x, y W A, 0, 1.21 where q is he Karlin McGregor deerminan 1.16. ii Le x = i=1 x2 i. Then Proof See Secion 3.3 in [24]. p, y 0 lim x 0 p, y x = 1/2 1 n=1 n! h y 2 p, y i 0, y W A, 0. 1.22 Denoe he -dimensional Laplacian wih respec o he variables x = x 1,..., x by i=1 2 x 2 i=1 i. Provided x, y W A, we can verify ha 1.21 saisfies he following parial differenial equaion PDE, p, y x = 1 2 p, y x + 1 i,j, i j 1 x i x j x i p, y x 1.23 wih he iniial condiion p 0, y x = δy x i=1 δy i x i. This PDE can be regarded as he backward Kolmogorov equaion of he sochasic process wih paricles, X = X 1,..., X, which solves he sysem of sochasic differenial equaions SDEs, dx i = db x i i + 1 j, j i d, 0, 1 i. 1.24 X i X j In 1962 Dyson [8] inroduced -paricle sysems of ineracing Brownian moions in R as a soluion X = X 1, X 2,..., X of he following sysem of SDEs: wih β > 0 and he condiion x = x 1, x 2,..., x W A for iniial posiions x i = X i 0, 1 i, dx i = db i + β 2 1 j, j i d X i X j, [0, T x, 1 i, 1.25 8

where {B i } i=1, 0 are independen BMs and T x ij = inf{ > 0 : X i = X j }, 1 i < j, T x = min ij. 1 i<j I is called Dyson s Brownian moion model wih parameer β [33, 11, 2, 1, 44]. We can prove ha, for any x W A, T x < if β < 1, and T x = if β 1 [38, 14]. 1 The presen sysem 1.24 is idenified wih Dyson s BM model wih parameer β = 2. We call his special case simply he Dyson model [43, 34, 13, 20]. Then he equivalence beween he Dyson model and he noncolliding Brownian moion is proved [13]. Theorem 1.3 The noncolliding Brownian moion X, 0 is equivalen in probabiliy law wih he Dyson model. Is ransiion probabiliy is given by ProbX dy Xs = x = p s, y xdy = h y h x de 1 i,j for 0 < s < <, x = x 1,..., x, y = y 1,..., y W A. and We can see ha [ p s, y i x j ] dy, 1.26 h x = 0. 1.27 h x > 0, if x W A, and h x = 0, if x W A. 1.28 Proposiion 1.2 i saes ha he Dyson model 1.26 is he h-ransformaion of he absorbing Brownian moion in W A [7, 13], in which he harmonic funcion is given by h x = x j x i. 1 i<j Therefore, a any posiive ime > 0 he configuraion is an elemen of W A, X = X 1, X 2,..., X W A, > 0, 1.29 and hence here are no muliple poins a which coincidence of paricle posiions, X i = X j, i j, occurs. We can consider, however, he Dyson model saring from iniial configuraions wih muliple poins. In order o describe configuraions wih muliple poins, we represen each paricle configuraion by a sum of dela measures in he form ξ = i I δ xi 1.30 1 The exisence of a srong and pahwise unique noncolliding soluion of SDEs 1.25 for general iniial condiions x 1 x 2 x was conjecured by Rogers and Shi [38]. I was proved by Cépa and Lépingle [5] using mulivalued SDE heory and by Graczyk and Ma lecki [15] by classical Iô calculus. See also [16, 17]. 9

wih a sequence of poins in R, x = x i i I, where I is a counable index se. Here for y R, δ y denoes he dela measure such ha δ y {x} = 1 for x = y and δ y {x} = 0 oherwise. Then, for 1.30 and A R, ξa = A ξdx = i I:x i A 1 = {x i, x i A}. The measures of he form 1.30 saisfying he condiion ξk < for any compac subse K R are called he nonnegaive ineger-valued Radon measures on R and we denoe he space hey form by M. The se of configuraions wihou muliple poins is denoed by M 0 = {ξ M : ξ{x} 1, x R}. There is a rivial correspondence beween W A and M 0. We call x R a labeled configuraion and ξ M an unlabeled configuraion. We consider he Dyson model as an M-valued diffusion process, Ξ, = saring from he iniial configuraion δ Xi, 0, 1.31 i=1 ξ = δ xi, 1.32 i=1 where X = X 1,, X is he soluion of 1.24 under he iniial configuraion x = x 1,..., x W A. We wrie he process as Ξ, Pξ and express he expecaion wih respec o he probabiliy law P ξ of he Dyson model by E ξ [ ]. We inroduce a filraion {F Ξ } [0, on he space of coninuous pahs C[0, M defined by F Ξ = σξs, s [0, ], where σ denoes he smalles σ-field. In order o characerize he process, we consider he Laplace ransformaions of he muliime join disribuion funcions of Ξ, P ξ. For any ineger M, a sequence of imes = 1,..., M [0, M wih 0 1 < < M <, and a sequence of funcions f = f 1,..., f M C c R M, le { Ψ M }] ξ [f] E [exp ξ f m xξ m, dx. 1.33 By 1.31, if we se es funcions as we can rewrie 1.33 in he form m=1 R χ m = e fm 1, 1 m M, 1.34 Ψ ξ [f] = E ξ [ M m=1 i=1 ] {1 + χ m X i m }. 1.35 We expand his wih respec o es funcions and define he spaio-emporal correlaion funcions {ρ ξ } as coefficiens, Ψ ξ [f] = 0 m, 1 m M M M m=1 W m A m=1 dx m m m i=1 χ m 10 x m i ρ ξ 1, x 1 1 ;... ; M, x M M, 1.36

= m i=1 dxm i, 1 m M. Here he empy producs equal 1 by convenion and he erm wih m = 0, 1 m M is considered o be 1. The funcion Ψ ξ [f] is a generaing funcion of correlaion funcions. where x m m denoes x m 1,..., x m m and dx m m 1.4 Fredholm deerminans and deerminanal processes Given an inegral kernel, Ks, x;, y, s, x,, y [0, R, and a sequence of funcions χ 1,..., χ M C c R M, M, he Fredholm deerminan associaed wih K and χ m M m=1 is defined as De s, { 1,..., M } 2, x,y R 2 = 0 m, 1 m M [ δ s δx y + Ks, x;, yχ y M M m=1 WA m m=1 dx m m m k=1 χ m x m k ] de 1 i m,1 j n, 1 m,n M [ K m, x m i ; n, x n j If we consider he simples case where M = 1 and 1 = [0, in 1.37, we have [ ] De δx y + K, x;, yχ y = x,y R 2 =0 W A dx k=1 ]. 1.37 χ x k de 1 i,j [K, x i;, x j ]. Given v = v 1,..., v W A, pu χ x = l=1 χ lδ vl x wih χ l R, 1 l. In his case he above is equal o =0 J I, J= k J χ k de i,j J [K ij] wih K ij = K, v i ;, v j, 1 i, j, where we wrie J I, J =, if J = {j 1,..., j }, 1 j 1 < < j. This is obained as he Fredholm expansion formula of de 1 i,j [δ ij + K ij χ j ], ha is, ] de [δ ij + K ij χ j = 1 i,j = 1 + + 1 i χ i K ii + 1 i<j<k =0 J I, J= k J 1 i<j χ i χ j χ k χ i χ j K ii K ji K ii K ij K ik K ji K jj K jk K ki K kj K kk χ k de i,j J [K ij] +. 1.38 For his reason, 1.37 is called he Fredholm deerminan. See, for insance, Chaper 21 in [33], Chaper 9 in [11], and Chaper 3 in [2] for more deails of Fredholm deerminans. 11 K ij K jj

Definiion 1.4 If any momen generaing funcion 1.33 is given by a Fredholm deerminan, he process Ξ, P ξ is said o be deerminanal. In his case all spaio-emporal correlaion funcions are given by deerminans as [ ] ρ ξ 1, x 1 1 ;... ; M, x M M = de K ξ m, x m i ; n, x n j, 1.39 1 i m,1 j n, 1 m,n M 0 1 < < M <, 1 m, x m m R m, 1 m M. Here he inegral kernel, K ξ : [0, R 2 R, is a funcion of he iniial configuraion ξ and is called he correlaion kernel. Remark 1.1 If he process Ξ, P ξ is deerminanal, hen, for each specified ime 0 <, all spaial correlaion funcions are given by deerminans as ρ ξ x = de 1 i,j [Kx i, x j ], 1, 1.40 wih Kx, y = K ξ, x;, y. In general a random ineger-valued Radon measure in M resp. M 0 is called a poin process resp. simple poin process. A simple poin process is said o be a deerminanal poin process DPP or Fermion poin process wih kernel K, if is spaial correlaion funcions exis and are given in he form 1.40. When K is symmeric, i.e., Kx, y = Ky, x, x, y R, Soshnikov [42] and Shirai and Takahashi [41] gave sufficien condiions for K o be a correlaion kernel of a deerminanal poin process see also [19, 2, 1]. The noion of deerminanal process given by Definiion 1.4 is a dynamical exension of he deerminanal poin process [3, 29]. The following has been esablished in [30, 23, 24]. Theorem 1.5 For any finie and fixed iniial configuraion ξ M, ξr =, he Dyson model is deerminanal. 2 Wigner s Semicircle Law as LL 2.1 Hermie orhonormal funcions and BM For BM, we perform he following ransformaion wih parameer α C {z = x + 1y : x, y R}, B ˇB α, ˇB α = which is called he Esscher ransformaion. I is easy o see ha E[e αb ] = eαb, 0, 2.1 E[e αb ] e αx p, x 0dx = e α2 /2, 0. 12

Then he above is wrien as ˇB α = G α, B, 0 wih For 0 < s <, G α, x = e αx α2 /2. 2.2 E[G α, B F s ] = E[eαB F s ] E[e αb ] = E[eαBs e αb Bs F s ]. E[e αbs e αb Bs ] By he definiion of F s and independence of incremen of BM he propery BM3, he numeraor is equal o e αbs E[e αb Bs ], and he denominaor is equal o E[e αbs ] E[e αb Bs ]. Hence he above equals e αbs /E[e αbs ] = G α s, Bs. This implies ha G α, B is a maringale: The funcion 2.2 is expanded as E[G α, B F s ] = G α s, Bs, 0 s. 2.3 G α, x = n=0 m n, x αn n! 2.4 wih m n, x = n/2 x H n, n 0 {0, 1, 2,... }. 2.5 2 2 Here {H n x} n 0 are he Hermie polynomials of degrees n 0, H n x = 1 n e x2 dn e x2 2.6 dx n [n/2] = 1 k n! k!n 2k! 2xn 2k, 2.7 where for a 0, [a] denoes he larges ineger which is no larger han a. k=0 Lemma 2.1 The funcions {m n, x} n 0 saisfy he following. i They are monic polynomials of degrees n 0 wih ime-dependen coefficiens: n 1 m n, x = x n + c k n x k, 0. k=0 13

ii For 0 k n 1, c k n 0 = 0. Tha is, m n 0, x = x n, n 0. iii If we se x = B, hey provide maringales: E[m n, B F s ] = m n s, Bs, 0 s, n 0. 2.8 Proof By he definiion 2.5 wih 2.7, i and ii are obvious. oe ha when n is even resp. odd, c k n 0 for odd resp. even k. Since G α, B, 0 was shown o be a maringale for any α C, he expansion 2.4 implies iii. Concerning he Hermie polynomials, we can prove he following. Exercises 1.2-1.4 in [24] See, for insance, Lemma 2.2 i The Hermie polynomials {H n x} n 0 have he orhogonaliy propery ii The formula is esablished. polynomials, e x2 H n xh m xdx = 2 n n! πδ nm, n, m 0. 2.9 n=0 H n z sn n! = e2sz s2 2.10 This implies he following conour inegral represenaions of he Hermie H n z = n! 2π dη e2ηz η2 1 Cδ 0 η, n 0, 2.11 n+1 where Cδ 0 is a closed conour on he complex plane C encircling he origin 0 once in he posiive direcion. iii The Hermie polynomials saisfy he following recurrence relaions: H n+1 z 2zH n z + 2nH n 1 z = 0, 2.12 H nz = 2nH n 1 z, n, 2.13 where H nz = dh n z/dz. From 2.12 and 2.13, he equaions H nz 2zH nz + 2nH n z = 0, n 0 are derived. Tha is, {H n z} n 0 saisfy he differenial equaion This is known as he Hermie differenial equaion. u 2zu + 2nu = 0. 2.14 14

By he orhogonaliy 2.9 if we pu φ n x = 1 π2n n! H nxe x2 /2, x R, n 0, 2.15 hen we have he equaliies, R dx φ n xφ m x = δ nm, n, m 0. 2.16 The funcions {φ n x} n 0 are called he Hermie orhonormal funcions on R. The following expression for he ransiion probabiliy densiy 1.1 of BM is known as Mehler s formula, /4s ps, x y = e x2 1 n/2 x y φ e y2 /4 n φ n, 0 < s <, x, y R. 2s s n=0 2s 2 2.17 See, for insance, Exercise 3.15 in [24] 2.2 Dyson model saring from ξ = δ 0 and Hermie kernel From now on, we will consider he Dyson model saring from he following special iniial configuraion, ξ = δ 0. 2.18 Tha is, he iniial configuraion ξ is he sae such ha all poins are concenraed on an origin. In his case, he correlaion kernel for he Dyson model is given by [34, 3, 29] K Hermie s, x;, y K δ 0 s, x;, y 1 1 2s s = n=0 1 2s n= n/2 x y φ n φ n 2s 2 s This kernel is called he exended Hermie kernel. Denoe he equal-ime correlaion kernel as n/2 x y φ n φ n 2s 2 for s, for s >. 2.19, K Hermie x, y K Hermie, x;, y 1 1 x y = φ n φ n, 0 < <. 2 2 2 n=0 15

By he recurrence formulas 2.12, we can see ha his has he following expression,, K Hermie x, y = 2, K Hermie x, x = 1 2 [ φ x/ 2φ 1 y/ 2 φ 1 x/ 2φ y/ 2, x y if x y, 2.20 { } 2 x φ ] x x + 1φ 1 φ +1, 2 2 2 2.21 0 < <. This spaial correlaion kernel is a special case of he Chrisoffel Dorboux kernel see, for insance, Chaper 9 in [11] and Chaper 3 in [2]. I is called he Hermie kernel and defines he deerminanal poin process [42, 41] on R such ha a spaial correlaion funcion is given by, ρ Hermie x = [ de 1 i,j ] Hermie x i, x j, K 2.22 for any 1 and x = x 1,..., x R, > 0. We wrie he probabiliy measure, of his DPP as P Hermie. 2.3 Wigner s semicircle law The densiy funcion a ime 0 and posiion x R is given by ρ Hermie, x = K, Hermie x, x = 1 1 2 x φ n 2 2 n=0 [ = 1 2 x φ ] x x + 1φ 1 φ +1. 2.23 2 2 2 2 I is easy o verify ha ρ Hermie, xdx = by he orhonormaliy 2.16 of {φ n x} n 0. We will obain esimaions for he asympoics a. The following formulas are derived from Theorem 8.22.9 a and b in Chaper VIII of [40]. Le ε and ω be a fixed 16

posiive numbers. We have 1/4 2 1 2 i φ + 1 cos ϕ = π sin ϕ { [ sin 2 + 1 sin 2ϕ 2ϕ + 34 ] 4 π 2 ii φ + 1 cosh ϕ = [ exp 2 + 1 4 } 1 + O, ε ϕ π ε 1/4 1 1 2π sinh ϕ 2 2ϕ sinh 2ϕ + 34 ] { π 1 + O 2.24 } 1, ε ϕ ω. 2.25 ow we apply hese asympoic esimaes o he Chrisoffel Darboux formula for he Hermie funcions, 1 n=0 2 2 φ n x = φ x + 1φ+1 xφ 1 x. 2.26 Case 1: x = 2 + 1 cos ϕ, ε ϕ π ε In order o calculae 2.26, firs we derive he asympoics for φ 1 x and φ +1 x from 2.24. We see 2 2 2 + 1 φ 1 + 1 cos ϕ = φ 1 1 + 1 2 1 cos ϕ where we have se = φ 1 2 1 + 1 cosϕ + η, 2.27 cosϕ + η = 2 + 1 cos ϕ. 2 1 Since 2 + 1 2 1 = 1 + 1 1 2 + O, 2 and we have cosϕ + η = cos ϕ η sin ϕ + Oη 2, η = 1 cos ϕ 1 2 sin ϕ + O. 2.28 2 17

Applying 2.24 o 2.27, we obain 1/4 2 1 2 φ 1 + 1 cos ϕ = π sin ϕ { [ 1 sin + 1 { } sin2ϕ + η 2ϕ + η + 2 4 3 ] } 1 4 π + O.2.29 Here, if we use 2.28, hen sin2ϕ + η 2ϕ + η sin 2ϕ 2ϕ + 2ηcos 2ϕ 1 sin 2ϕ 2ϕ 1 cos ϕ cos 2ϕ 1 sin ϕ = sin 2ϕ 2ϕ + 2 sin ϕ cos ϕ = sin 2ϕ 2ϕ + 1 sin 2ϕ, and hence 1 2 + 1 { } sin2ϕ + η 2ϕ + η 4 2 + 1 1 sin 2ϕ 2ϕ + ϕ + O. 4 Therefore, by 2.29, we have he esimae Similarly, we pu wih Then 2 φ 1 + 1 cos ϕ = { sin φ +1 2 + 1 cos ϕ and we obain he esimae [ 2 + 1 4 2 φ +1 + 1 cos ϕ = { sin 1/4 1 2 π sin ϕ { sin 2ϕ 2ϕ} + 34 ] π + ϕ } 1 + O. 2.30 2 2 + 1 = φ +1 + 1 + 1 2 + 3 cos ϕ = φ +1 2 + 1 + 1 cosϕ + η, 2.31 2 + 1 cosϕ + η = cos ϕ. 2 + 3 η = 1 cos ϕ 1 2 sin ϕ + O, 2 [ 2 + 1 4 1/4 1 2 π sin ϕ { sin 2ϕ 2ϕ} + 34 ] π ϕ 18 } 1 + O. 2.32

Insering hem ino 2.26 gives 1 n=0 φ n 2 + 1 cos ϕ 2 = { [ 2 sin 2 π sin ϕ 2 + 1 sin 2ϕ 2ϕ + 34 ] 4 π [ sin sin 2ϕ 2ϕ + 34 ] π + ϕ sin 1 +O. 2 + 1 4 [ 2 + 1 4 sin 2ϕ 2ϕ + 34 π ϕ ]} Pu A = 2 + 1 sin 2ϕ 2ϕ + 3 4 4 π. Then we see { } = sin 2 A sina + ϕ sina ϕ = sin 2 A sin A cos ϕ + cos A sin ϕsin A cos ϕ cos A sin ϕ = sin 2 A sin 2 A cos 2 ϕ cos 2 A sin 2 ϕ = sin 2 A1 cos 2 ϕ + cos 2 A sin 2 ϕ = sin 2 A sin 2 ϕ + cos 2 A sin 2 ϕ = sin 2 ϕ and hence 1 n=0 φ n 2 + 1 cos ϕ 2 = = 2 π sin ϕ sin2 ϕ + O 2 sin ϕ + O π 1 1. 2.33 Since sin ϕ = 1 cos 2 ϕ = 1 2 2 x2, 1 x2 2 we obain in, 1 n=0 φ n x 2 1 2 x2, 2 < x < 2. π 19

Case 2: x = 2 + 1 cosh ϕ, ε ϕ ω We pu φ 1 2 + 1 cosh ϕ 2 2 + 1 = φ 1 1 + 1 2 1 cosh ϕ = φ 1 2 1 1 coshϕ + η, where coshϕ + η = 2 + 1 coshϕ + η 2 1 1 + 1 1 2 + O cosh ϕ. 2 Since we have Inser he above ino 2.25. Then we have For coshϕ + η = cosh ϕ + η sinh ϕ + Oη 2, η = 1 cosh η 1 2 sinh η + O. 2 2 1/4 1 1 φ 1 + 1 cosh ϕ = 2π sinh ϕ 2 [ 1 exp + 1 { 2ϕ + η sinh 2ϕ + η} 2 4 ] { } 1 1 + O. 2ϕ + η sinh 2ϕ + η 2ϕ sinh 2ϕ + 2η1 cosh 2ϕ 2ϕ sinh 2ϕ + 1 cosh ϕ sinh ϕ 2 sinh2 ϕ = 2ϕ sinh 2ϕ 1 2 sinh ϕ cosh ϕ = 2ϕ sinh 2ϕ 1 sinh 2ϕ, he above is wrien as 2 φ 1 + 1 cosh ϕ = [ exp 2 + 1 4 1 1 2π sinh ϕ 2 2ϕ sinh 2ϕ ϕ 1/4 ] { 1 + O } 1. 20

Similarly, we obain 2 φ +1 + 1 cosh ϕ = [ exp 2 + 1 4 1 1 2π sinh ϕ 2 2ϕ sinh 2ϕ + ϕ 1/4 ] { 1 + O Insering he above ino 2.26 implies ha, in he limi, 1 ϕ n 2 + 1 cosh ϕ 2 } 1. n=0 1 { exp [ + 1/22ϕ sinh 2ϕ] 2π sinh ϕ 2 exp [/2 + 1/42ϕ sinh 2ϕ ϕ] exp [/2 + 1/42ϕ sinh 2ϕ + ϕ] } { 1 + O 1 1 2π sinh ϕ 2 O = O 1/2 0. } 1 Using he above evaluaions, we have he asympoics of he densiy profile a, 1 ρ Hermie, x π 2 x2 2 2, if 2 x 2, 2.34 0, oherwise. The disribuion of paricles has a finie suppor, whose inerval, and hus ρ Hermie, x as for fixed 0 < <, when x 2, 2. If we se x = 2 ξ, we see ha lim 1 ρ Hermie, 2 ξdx = 2 1 ξ2 dξ, if 1 ξ 1, π 0, oherwise, 2.35 which is known as Wigner s semicircle law [33]. Here we regard his as he law of large numbers LL for P, Hermie wih fixed 0,. See, for insance, Secion 5 in [2] for he imporance of Wigner s semicircle law in free probabiliy. 21

3 Scaling Limis and Infinie Paricle Sysems 3.1 Bulk scaling limi and sine kernel Firs we consider he cenral region x 0 in he semicircle-shaped profile of paricle densiy in he scaling limi. 3.1 π2 In his limi he sysem becomes homogeneous also in space wih a consan densiy ρ = 1. We call his he bulk scaling limi. Proposiion 3.1 For any M, any sequence { m } M m=1 of posiive inegers, and any sricly increasing sequence {s m } M m=1 of posiive numbers, lim ρ Hermie π + 2s 1, x 1 2 1 ; ; π + 2s M, x M 2 M [ ] = de 1 i m,1 j n, 1 m,n M K sin s m, x m i ; s n, x n j ρ sin s 1, x 1 1 ; ; s M, x M M, 3.2 where K sin s, x;, y = 1 0 du e π2 u 2 s cos{πux y}, if > s, K sin x, y, if = s, 3.3 wih Proof K sin x, y = 1 0 For any u R, he formulas 1 du e π2 u 2 s cos{πux y}, if < s, du cos{πux y} = lim l 1l l 1/4 φ 2l lim l 1l l 1/4 φ 2l+1 sin{πx y}, x, y R. 3.4 πx y u 2 = 1 cos u, l π u 2 l = 1 π sin u 3.5 22

are known see Eq. 8.22.8 in Chaper VIII of [40]. We noe ha n α = /π 2 + 2s n α m = /π 2 + 2s m { } α/ 1 + 2π2 s n 1 + 2π2 s m e 2π2 αs m s n/ for 1 wih a fixed number α. Then 2.19 wih s = m = /π 2 + 2s m = n = /π 2 + 2s n is evaluaed a as K Hermie m, x; n, y 1 [/2 1] e 2π2 ls m s n/ l=0 { } 2l 2l 2l 2l cos π 2l x cos π y + sin π x sin π y 1 1 dλ { e π2 λs m s n cosπ λx cosπ λy + sinπ λx sinπ } λy 2 λ = 0 1 0 du e π2 u 2 s m s n cos{πux y}. In paricular, when m = n, i.e., s n s m = 0, he inegraion is readily performed o have 1 du cos{πux y} = sin{πx y}/πx y. A similar evaluaion a can be 0 done also for 2.19 wih s = m > = n. Remark 3.1 The correlaion kernel 3.3 is called he exended sine kernel. Since i is a funcion of s and x y, he deerminanal process obained by he bulk scaling limi is a emporally and spaially homogeneous process wih an infinie number of paricles, which we wrie as Ξ, P sin. Le P sin be a saionary probabiliy measure on R, which is a deerminanal poin process [42, 41] such ha he spaial correlaion funcion is given by [ ] ρ sin x = K sin x i, x j 3.6 de 1 i,j for any, x = x 1,..., x R, where K sin is given by 3.4. The deerminanal process Ξ, P sin is reversible wih respec o P sin. 3.2 Sof-edge scaling limi and Airy kernel ex we consider he scaling limi 1/3 and x 2 2/3. 3.7 23

Since 3.7 gives x 2 /2 2, his scaling limi allows us o zoom ino he righ edge of he semicircle-shaped profile 2.34, and we obain a spaially inhomogeneous infinie paricle sysem. Following random marix heory [33], we call 3.7 he sof-edge scaling limi. In order o describe he limi, we inroduce he Airy funcion Aix = 1 k 3 dk cos π 3 + kx. 3.8 I is he soluion of he equaion which obeys he asympoics given by 1 Aix 2 exp 23 πx1/4 x3/2, 1 2 Ai x cos πx 1/4 3 x3/2 π 4 0 d 2 Aix = xaix, 3.9 dx2 as x. 3.10 In he proof of he following heorem, we will use he formula 2l lim l 2 1/4 l 1/12 x φ l + 2 l 1/6 = Aix for x R, 3.11 which is obained from Theorem 8.22.9 c in Chaper VIII of [40]. Le a s = 2 2/3 + 1/3 s s2 4, 3.12 and a s + x = a s + x 1, a s + x 2,, a s + x. Proposiion 3.2 For any M, any sequence { m } M m=1 of posiive inegers, and any sricly increasing sequence {s m } M m=1 of posiive numbers lim ρ Hermie 1/3 + s 1, a s 1 + x 1 1 ; ; 1/3 + s M, a s M + x M M [ ] = de K Airy s m, x m i ; s n, x n j 1 i m,1 j n, 1 m,n M ρ Airy s 1, x 1 1 ; ; s M, x M M, 3.13 where K Airy s, x;, y = 0 0 du e us /2 Aix + uaiy + u, if s, du e us /2 Aix + uaiy + u, if < s. 3.14 24

Proof have Replacing he summaion index in 2.19 by p 1 for he case where m n, we K Hermie m, x; n, y 1/2 1 n 1 = 2m m p=0 When we se m = 1/3 + s m, we see ha n m p/2 x y φ p 1 φ p 1. 2m 2n a s m + x 2m = 2 + x 2 1/6 + O 1/2, 3.15 and we can use he formula 3.11: For n a s m + x 2 x φ p 1 φ p 1 + 2 1/6 2m φ p 1 2 p 1 + 1 2 p 1 1/6 { 2 1/4 1/12 Ai x + p. 1/3 x + p } 1/3 [ p/2 ] 1 + sn / 1/3 1/3 p/ /2 1/3 = e psm sn/2 1/3 as, m we have, for n m, 1 + s m / 1/3 K Hermie 1/3 + s m, a s m + x; 1/3 + s n, a s n + y 1 1 e psm sn/2 1/3 Ai x + p Ai y + 1/3 1/3 p=0 0 p 1/3 du e usm sn/2 Aix + uaiy + u as. oe ha he facor n / m 1/2 was omied in he second line in he above equaions, since i is irrelevan in calculaing deerminans. A similar evaluaion a of 2.19 can be done also for m > n. The infinie sysem obained by he sof-edge scaling limi 3.7 is emporally homogeneous, bu spaially inhomogeneous as shown by he correlaion kernel K Airy, 3.14. We call K Airy he exended Airy kernel [35, 25] and wrie his saionary deerminanal process as Ξ, P Airy. Prähofer and Spohn [36] and Johansson [21] sudied he righmos pah in he presen sysem and called i he Airy process A 0. 25

Remark 3.2 Le P Airy be he saionary probabiliy measure on R, which is a deerminanal poin process [42, 41] such ha he spaial correlaion funcion is given by [ ] ρ Airy x = K Airy x i, x j 3.16 de 1 i,j for any, x = x 1,..., x R, where The Airy kernel K Airy is also wrien as where Ai x = daix/dx. K Airy x, y = K Airy, x;, y = 0 du Aix + uaiy + u. 3.17 K Airy x, y = AixAi y Ai xaiy, x y, 3.18 x y K Airy x, x = Ai x 2 xaix 2, 3.19 4 Tracy Widom Disribuion as CLT 4.1 Disribuion funcion of he maximum posiion of paricles Consider a deerminanal poin process Ξ, P wih an infinie number of paricles on R such ha Ξ = i δ X i M 0. Assume ha he correlaion kernel is given by Kx, y, x, y R 2 [42, 41]. Two examples, Ξ, P sin wih he sine kernel K sin and Ξ, P Airy wih he Airy kernel K Airy, were given in he previous secion. Wih a es funcion χ C c R he generaing funcion of spaial correlaion funcions defined by [ ] Ψ[χ] = E {1 + χx i }, 4.1 i is expressed by a Fredholm deerminan [ ] Ψ[χ] = De δx y + Kx, yχy. 4.2 x,y R 2 Le 1 ω be he indicaor funcion of condiion ω; 1 ω = 1 if ω is saisfied, and 1 ω = 0 oherwise. If we se χx = 1 x s wih a parameer s R, 4.1 becomes [ ] [ ] Ψ[ 1 s ] = E {1 1 Xi s} = E i [ ] = P X i < s, i 26 i 1 Xi <s [ = P max X i < s i ].

This is he disribuion funcion of he maximum posiion of paricles, and by 4.2, i has he Fredholm deerminanal expression [ ] [ ] P max X i < s = De δx y K s x, y, 4.3 i x,y R 2 where K s x, y = Kx, y1 y s, x, y, s R. 4.4 For inegrable funcions f i x, y, i, x, y R 2, we use he following noaions, [f 1 f 2 f n ]x 1, x n+1 = f 1 x 1, x 2 f 2 x 2, x 3 f n x n, x n+1 dx 2... dx n, R n 1 n {2, 3,... }, x 1, x n+1 R. We regard f 1 f 2 f n as an operaor such ha is x, y-elemen is given by [f 1 f 2 f n ]x, y, x, y R 2. The race of an operaor f is defined by Trf = fx, xdx, 4.5 R and if Trf <, f is said o be a race class operaor [39]. Pu 1x, y = δx y, x, y R 2. The resolven of K a is defined by Le and ρ a = K n a 1 K a 1. 4.6 n=0 R a ρ a K a = n=0 K n+1 a, 4.7 ra = R a a, a lim R a x, y, a R. 4.8 y x x=a The correlaion kernels K of deerminanal poin processes are race class operaors and he following exponenial expression for 4.3 is proved see, for insance, Lemmas 2.1 and 2.2 in [41]. Lemma 4.1 If r is inegrable, [ ] P max X i < s i = exp s da ra, s R. 4.9 Proof The explici expression of 4.3 is [ ] De δ x {y} K s x, y = 1 + x,y R 2 1 n I n s n! n=1 27

wih I n s = dx n R n Using he Maclaurin expansion we can show ha n=1 log We rewrie his as 1 Tr n Kn s = = 1 + log1 x = n=1 de [K sx i, x j ]. 1 i,j n n=1 x n n, 4.10 1 n 1 I n s = n! n Tr Kn s. 4.11 s s da a Tr da Tr n=1 n=1 1 n Kn a K n 1 K a a a n=1 = s K a da Tr ρ a. a Here K a a x, y = { } Kx, y1y a a = Kx, y a 1 y a = Kx, yδy a, and hence The proof is compleed. K a Tr ρ a a [ ] K a = dx ρ a x, x R a = dx dy ρ a x, y K a y, x R R a = dy ρ a a, yky, a = R a a, a. R For x, y R 2, as a, K a x, y = Kx, y1 y a 0. Then he definiion 4.7 gives R a x, y K a x, y = Kx, y1 y a in a for x, y R 2. If we pu x = y = a, we have ra Ka, a as a. 4.12 oe ha, by definiions 4.6 and 4.7, ρ a = 1 + R a and hence ρ a x, y = δx y + R a x, y, x, y R 2. 4.13 28

4.2 Inegrals involving resolven of correlaion kernel Le, 0,. ow we assume The formula 2.20 is simply wrien as wih Ax = 2 We can prove he following., Kx, y = K Hermie x, y, x, y R2. 4.14 Kx, y = AxBy BxAy, x y, 4.15 x y 1/4 x φ, Bx = 2 1/4 x φ 1. 4.16 2 2 Lemma 4.2 Le Then P a x = Q a x = R R dz ρ a x, zbz = [ρ a B]x, dz ρ a x, zaz = [ρ a A]x. 4.17 [ dqa x ra = dx P ax dp ] ax dx Q ax. 4.18 x=a Using 4.16 and 4.17, we define he following inegrals, wa = dx P a x1 x a Bx = dx P a xbx, 4.19 R a ua = dx Q a x1 x a Ax = dx Q a xax. 4.20 Then he following is proved. R Lemma 4.3 The following equaions hold: dp a x = x dx 2 P ax + wa Q a x + R a x, ap a a, dq a x = x dx 2 Q ax + ua P a x + R a x, aq a a, 4.21 and R a x, x = { x P axq a x + ua P a x 2 + + wa Q a x 2 } + R a x, a{q a ap a x Q a xp a a} a 1 x a. 4.22 29

4.3 onlinear hird-order differenial equaion Le By he definiion 4.8, 4.22 gives Is derivaive is ra = a paqa + ua pa = P a a, qa = Q a a, a R. 4.23 pa 2 + r a = 1 paqa a p aqa a paq a u a pa 2 + 2 ua pap a + w a qa 2 + 2 + wa We find he following sysem of differenial equaions. Lemma 4.4 For a R, p a = a 2 pa + wa q a = a 2 qa + ua + wa qa 2. 4.24 qaq a. 4.25 qa, 4.26 pa, 4.27 w a = pa 2, 4.28 u a = qa 2. 4.29 Insering 4.26 4.29 ino 4.25 gives a remarkably simple equaion, r a = 1 paqa. 4.30 Moreover, Tracy and Widom derived he following resul [45, 46]. Proposiion 4.5 The funcion ra solves he following nonlinear hird-order differenial equaion, a r 2 a 4 r a + a 2 ra + 2 6r a 2 = 0. 4.31 30

Proof By 4.26 and 4.27, we have paqa = p aqa + paq a = ua pa 2 + wa qa 2. 4.32 On he oher hand, 4.28 and 4.29 give ua wa + 1 uawa = u a w a + 1 u awa + uaw a = ua pa 2 + wa qa 2. 4.33 Then, we find he equaliy paqa = ua wa + 1 uawa. 4.34 For finie x, 1 x a 0 as a, and Aa 0, Ba 0 as a. Therefore pa, qa, wa and ua all become zero as a. By inegraing boh sides of 4.34 from a o, we obain he equaliy paqa = ua wa + 1 uawa. 4.35 If we use 4.32, he derivaive of 4.30 is wrien as { } r a = 1 ua pa 2 + wa qa 2, and hen { } r a = 2 2 pa2 qa 2 a 2 ua pa 2 + + wa qa 2 [ { }] + 4 paqa 1 ua wa + 1 uawa, 4.36 where 4.26 4.29 are used. By 4.24 and 4.35, 4.36 is rewrien as r a = a a 2 ra 2 4 paqa 6 3 2 2 paqa2. By combining i wih 4.30, we obain 4.31. This complees he proof. 31

4.4 Sof-edge scaling limi For, 0,, we perform he variable ransformaion a u by a = 2 + 1/6 u u = a 2 1/2 1/6. 4.37 Since / a = 1/2 1/6 / u, if we se ru = 1/2 1/6 ra wih 4.37, 4.31 is ransformed ino r u 4u r u + 2 ru + 6 r u 2 2/3 u{u r u ru} = 0. On he oher hand, 4.9 is wrien as exp du ru. s 2 1/2 1/6 Le x = s 2 1/2 1/6. Then max X i < s 1 i max X i 2 1 i < x. 1/2 1/6 Therefore, we have he following limi, lim P, Hermie max X i 2 1 i 1/2 1/6 < x = exp x du ru, 4.38 where ru solves he equaion r u 4u r u + 2 ru + 6 r u 2 = 0. 4.39 Wih 4.37, he BM scaling variable a/ 2 behaves as a = 2 + 1 1/6 u, 2 2 which is he same as 3.15. Then he presen limi realizes he sof-edge scaling limi discussed in Secion 3.2. By Proposiion 3.2, we can conclude ha he lef-hand side of 4.38 is equal o [10] [ ] P Airy max X i < x 1 i [ = De δ u {v} K Airy u, v1 v x ], x R, u,v R 2 where he Airy kernel, K Airy, is given by 3.18. Since P Airy is a saionary probabiliy measure, his disribuion obained in he limi 4.38 does no depend on ime 0,. 32

4.5 Painlevé II and limi heorem of Tracy and Widom Le Then 4.39 is wrien as ru = f u 2 + fuf u 2ufu 2 u dv fv 2. 4.40 u dv fv 2 3fu 4 = 0. If we differeniae his equaion by u, we obain { fu d } { } du + 3f u f u ufu 2fu 3 = 0. Here we consider he equaion f u = ufu + 2fu 3, 4.41 which is a special case of he Painlevé II equaion see, for insance, Chaper 21 and Appendix A.45 in [33], Chaper 8 in [11], Chaper 3 in [2], and Chaper 9 in [1]. Since 4.40 gives x du ru = = x x du u dv fv 2 v dv fv 2 du = x he RHS of 4.38 is wrien as exp x dv v xfv2. By 4.12, in he sof-scaling limi, we find ru K Airy u, u as u. We noe ha he inegral represenaion of K Airy 3.17 gives K Airy u, u = 0 dw Aiu + w 2 = Comparing his wih 4.40, we can conclude ha x u dv v xfv 2, dv Aiv 2. fu Aiu as u. 4.42 Hasings and McLeod [18] proved ha he Painlevé II equaion 4.41 has a unique soluion f HM u, which saisfies 4.42. ow we arrive a he following limi heorem for he maximum posiion of paricles of he Dyson model wih an infinie number of paricles. 33

0.5 0.4 0.3 0.2 0.1 6 4 2 0 2 4 Figure 1: The probabiliy densiy funcion of he Tracy Widom disribuion 4.46 is shown by a red curve. The black curve shows he probabiliy densiy funcion of he Gaussian disribuion 4.48 wih he same values of mean and variance as he Tracy Widom disribuion given by 4.47 µ = µ TW, σ 2 = σ 2 TW. Theorem 4.6 For any 0,, he probabiliy lim P, Hermie has he following wo expressions, max X i 2 ] 1 i < x = P 1/2 1/6 Airy [max X i < x, x R, 4.43 i [ ] F TW x = De δ u {v} K Airy u, v1 v x 4.44 u,v R 2 = exp dv v xf HM v 2, x R. 4.45 x Here he former is he Fredholm deerminanal expression, and he laer is he expression in erms of he Hasings McLeod soluion f HM of he Painlevé II equaion 4.41. We regard 4.43 as he cenral limi heorem CLT of P, Hermie in a he righ sof edge in which he mean value is given by 2. The exponen of he CLT is 1/6, which is very differen from he classical exponen 1/2 in he Gaussian classical CLT. The probabiliy disribuion funcion 4.45 is called he Tracy Widom disribuion [45, 46]. I has he probabiliy densiy funcion p TW x = df TWx, x R. 4.46 dx umerical values of he mean, variance, skewness, and kurosis are he following see [47], 34

1 0.100 0.010 0.001 0.0001 6 4 2 2 4 Figure 2: The semi-log plos of he probabiliy densiy funcion of he Tracy Widom disribuion 4.46 he red curve and ha of he Gaussian disribuion 4.48 wih he same values of mean and variance he black curve. Secion 9.4.2 in [11], and [37], µ TW = σ 2 TW = xp TW xdx = 1.771086807, x µ TW 2 p TW xdx = 0.813194792, S TW = K TW = 3 x µtw p TWxdx = 0.224084203, σ TW 4 x µtw p TWxdx 3 = 0.093448087. 4.47 σ TW Figure 1 shows he comparison beween p TW x and he probabiliy densiy funcion of he Gaussian disribuion p G x = 1 2πσ e x µ2 /2σ 2, x R, 4.48 wih he same values of mean and variance as he Tracy Widom disribuion given by 4.47 µ = µ TW, σ 2 = σ 2 TW. The difference beween p TW and p G can be shown beer, if we represen hem in he semi-log plos as given by Fig. 2. Acknowledgemens This manuscrip was prepared for Mini Workshop : Modern Theory of Sochasic Paricles in 27-28 June 2018 a Polyechnic of Wroc law. The auhor expresses his graiude o he organizers, Jacek Ma lecki and Pior Graczyk. The auhor is on sabbaical leave from Chuo Universiy, and his manuscrip was prepared a Fakulä für Mahemaik, Universiä Wien, 35

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