From the Pearcey to the Airy process

Transcription

1 From the Pearcey to the Airy process arxiv:19.683v1 [math.pr] 3 Sep 1 Mark Adler 1, Mattia Cafasso, Pierre van Moerbeke 3 Abstract Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian motions on the real line for the eigenvalues, as was discovered by Dyson. Applying scaling limits to the random matrix models, combined with Dyson s dynamics, then leads to interesting, infinite-dimensional diffusions for the eigenvalues. This paper studies the relationship between two of the models, namely the Airy and Pearcey processes and more precisely shows how to approximate the multi-time statistics for the Pearcey process by the one of the Airy process with the help of a PDE governing the gap probabilities for the Pearcey process. 1 Introduction Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian motions on R for the eigenvalues, as was discovered by Dyson [15]. Applying scaling limits to the random matrix models, combined with Dyson s dynamics, then leads to interesting, infinite-dimensional diffusions for the eigenvalues. This paper studies the relationship between two of the models, namely the Airy and Pearcey processes and more precisely shows how to approximate the Pearcey process by the Airy process with the help of a PDE [] governing the gap probabilities for the Pearcey process. The Airy process was introduced by Prähofer-Spohn [] and further developed by K. Johansson [17, 18]. A simple non-linear 3rd order PDE for the transition probabilities for this process was found in [3]; see also [3, 4]. The Pearcey process was introduced in [5, 1] in the context of non-intersecting Brownian motions and plane partitions, also based on prior work on matrix models with external source [, 1, 6, 7, 11, 13, 5, 7, 8, 9]. 1 Department of Mathematics, Brandeis University, Waltham, Mass 454, USA, adler@brandeis.edu. The support of a National Science Foundation grant # DMS is gratefully acknowledged Centre de recherches mathématiques, Université de Montréal; C. P. 618, succ. centre ville, Montréal, Québec, Canada H3C 3J7 Department of Mathematics and Statistics, Concordia University; 1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8, cafasso@crm.umontreal.ca 3 Département de Mathématiques, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis1University, Waltham, Mass 454, USA, pierre.vanmoerbeke@uclouvain.be. The support of a National Science Foundation grant # DMS-7-471, a European Science Foundation grant MISGAM, a Marie Curie Grant ENIGMA, Nato, FNRS and Francqui Foundation grants is gratefully acknowledged.

2 Consider n nonintersecting Brownian particles on the real line R, < x 1 t <... < x n t < with local Brownian transition probability given by pt; x, y := 1 πt e y x t, all starting from the origin x = at time t = and such that n particles are forced to ± n at t = 1. For very large n, the average mean density of particles has its support, for t 1, on one interval centered about x =, and for 1 < t 1 on two intervals symmetrically located about x =. The end points of the intervals of support describe a heart-shaped region in x, t-space, with a cusp at x, t =, 1/. The Pearcey process Pτ is defined see Figure 1 as the motion of these nonintersecting Brownian motions for large n, about x, t =, 1/ i.e., near the cusp, with space microscopically rescaled by a factor of n 1/4 and time rescaled by a factor n 1/, in tune with the Brownian motion rescaling. A partial differential equation for the Pearcey process was found by Adler-van Moerbeke [4] and in [] a much better version was obtained, namely a simple third order nonlinear PDE for the transition probabilities; it was obtained by a scaling limit on a PDE for non-intersecting Brownian motions with target points. This PDE is related to the Boussinesq equation and its hierarchy; this is part of a general result on integrable kernels, as explained in the forthcoming paper [1]. Near the boundary of the heart-shaped region of Figure 1, but away from the cusp, the local fluctuations behave as the so-called Airy process, which describe the non-intersecting brownian motions with space stretched by the customary GUE edge rescaling n 1/6 and time rescaled by the factor n 1/3, again in tune with the Brownian motion space-time rescaling. This paper shows how the Pearcey process statistics tends to the Airy process statistics when one is moving out of the cusp x = 3t t 7 3/ very near the boundary, that is at a distance of 3τ 1/6 for τ very large, with τ being the Pearcey time. To be precise, in the two-time case, the times τ i must be sufficiently near -in a very precise way- for the limit to hold. The main result of the paper can be summarized as follows:

3 Theorem 1.1 Given finite parameters t 1 < t, let both τ 1, τ, such that τ τ 1 behaves in the following precise way: τ τ 1 t t 1 = 3τ 1 1/3 + t t 1 3τ 1 + t 1t + O τ 5/3 1/ τ 1 The parameters t 1 and t provide the Airy times in the following approximation of the Airy process by the Pearcey process: { Pτi P 3τ } 7 i 3/ E 3τ i=1 i 1/6 i = = P {At i E i = } 1 + O τ 4/3 1 i=1 The same estimate holds as well for the one-time case. A similar but different result was then obtained in [6] for the one time case in the situation when one is moving out of the cusp following both the two branches of the cusp simultaneously. The proof of this Theorem proceeds in two steps. At first, Proposition.1, stated later, shows that, using the scaling above, the Pearcey kernel tends to the Airy kernel. In a second step, we show, using the PDE for the Pearcey process [] and Proposition.1, the result of Theorem 1.1. It is a fact that both the Airy and Pearcey processes are determinantal processes, for which the multi-time gap probabilities is given by the matrix Fredholm determinant of the matrix kernel, which will be described here. The Airy process is a determinantal process, for which the multi-time gap probabilities 4 are given by the matrix Fredholm determinant for t 1 <... < t l, l P At i E i = i=1 = det I [χ Ei Kt A i t j χ Ej ] 1 i,j l 1.3 of the matrix kernel in x = x 1,..., x l and y = y 1,..., y l, denoted as 4 χ E is the indicator function for the set E 3

4 follows: K A t 1,...,t l x, y d x d y := K A t 1,t 1 x 1, y 1 dx 1 dy 1... K A t 1,t l x 1, y l dx 1 dy l. Kt A l,t 1 x l, y 1 dx l dy Kt A l,t l x l, y l dx l dy l, 1.4 where, for arbitrary t i and t j, the extended Airy kernel K A t i,t j x, y is given by see Johansson [17, 18] where K A t i,t j x, y := K A t i,t j x, y 1It i < t j p A t j t i, x, y, 1.5 K A t i,t j x, y := = 1 πi e λt i t j Ax + λay + λdλ Γ > dv Γ< du e v3 /3+yv e u3 /3+xu p A t, x, y := 1 4πt e t3 1 x y 4t t y+x, for t >. 1 v + t j u + t i The contour u Γ < consists of two rays emanating from the origin with angles θ 1 and θ 1 with the positive real axis, and the contour v Γ > also consists of two rays with angles θ and θ with the negative real axis, as indicated in Figure. As is well known, one may choose θ 1, θ, θ 1, θ = π/3. In particular, for t i = t j, this is the customary Airy kernel K A t i,t i := K A x, y := dλax + λay + λ = AxA y AyA x. x y 1.6 4

5 n/ # of particles n/ - n/ t n/ t = 1 t = 1/ x Figure 1: The Pearcey process. V Y U X U X 1 σ σ τ 1 τ σ σ 1 U X U X 1 V Y π/8 < σ 1, σ, σ 1, σ < 3π/8 v Γ > u Γ < θ θ 1 θ θ 1 v Γ > u Γ < π/6 < θ 1, θ, θ 1, θ < π/ 3π/8 < τ, τ < 5π/8 Pearcey contour Airy contour Figure : Integration paths for the kernels. 5

6 The Pearcey process is determinantal as well, for which the multi-time gap probability is given by the Fredholm determinant for τ 1 <... < τ l, l P Pτ i E i = = det 1I [χ Ei Kτ P i τ j χ Ej ] i,j l i=1 for the Pearcey matrix kernel in ξ = ξ 1,..., ξ l and η = η 1,..., η l, denoted as follows: K P τ 1,...,τ l ξ, η dξ d η Kτ P 1,τ 1 ξ 1, η 1 dξ 1 dη 1... Kτ P 1,τ l ξ 1, η l dξ 1 dη l :=.. Kτ P l,τ 1 ξ l, η 1 dξ l dη 1... Kτ P l,τ l ξ l, η l, dξ l dη l 1.8 where for arbitrary τ i and τ j, see Tracy-Widom [5] K P τ i,τ j ξ, η = K P τ i,τ j ξ, η 1Iτ i < τ j p P τ j τ i ; ξ, η, 1.9 with K τ P i,τ j ξ, η := 1 du 4π X Y dv V U p P τ; ξ, η := 1 πτ e ξ η τ, for τ >, e V τ j V V η e U4 4 + τ i U Uξ where X and Y are the following contours: X = X 1 X consists of four rays emanating from the origin with angles σ 1, σ 1 with the positive real axis and σ, σ with the negative real axis, as given in Figure. The contour Y consists of two rays emanating from the origin with angles τ, τ with the negative real axis; it is customary to pick σ 1 = σ = σ 1 = σ = π/4 and τ = τ = π/. In [4, ], it was shown that, given intervals E i = ξ i 1, ξ i, the log of the probability l Q = Qτ 1,..., τ l ; E 1,..., E l := log P {Pτ i E i = } 6 i=1

7 satisfies a non-linear PDE, which we describe here. Given the times and the intervals above, one defines the following operators: τ := i ε τ := i τ i, Ei := k τ i τ i, ξ i k ε E := i, E := i k ξ i k ξ i k Ei. 1.1 Then Q satisfies the Pearcey partial differential equation in its arguments and the boundary of the intervals: 3 τ Q+ 1 4 ε τ +ε E EQ i τ i Ei τ E Q+ { τ E Q, EQ } E = From the Pearcey to the Airy kernel Define the rational functions Φx, t; u := 1 43u 3 hx, t; u := ux 4 x + 6t, t 3u + x t 3u tx + u 6 t x and the diagonal matrix e Φx,t 1 ;z 4 hx,t 1 ;z 4 S := e Φx,t ;z 4 hx,t ;z 4..1 We now state: Proposition.1 Given a parameter z, define two times τ 1 and τ blowing up like z 6 and depending on two parameters t 1 and t, τ i = 1 3z t iz 4 + Oz 1.. Given large ξ i and η j, define new space variables x i and y j, using τ i above, ξ i = 7 3τ i 3/ 3τ i 1/6 x i η j = 7 3τ j 3/ 3τ j 1/6 y j..3 7

8 With this space-time rescaling, the following asymptotic expansion holds for the Pearcey kernel in powers of z 4, with polynomial coefficients in t 1 + t, S K P τ 1,τ ξ, η dξd η S 1 = 1 + t 1 +t O 1 z 4 K A t 1,t x, y dxdy + O z 8,.4 where O 1 refers to a differential operator in / x, / y, with polynomial coefficients in x, y, t 1 ± t and t 1 t 1 and varying with the four matrix entries of the matrix kernel. On general grounds, due to the Fredholm determinant formula, this would give Theorem 1.1, but with a much poorer estimate in 1., namely instead of Oτ 4/3 1, the estimate would be Oτ /3 1. The existence of the PDE for the Fredholm determinant of the Pearcey process enables us to bootstrap the Oτ /3 1 estimate to Oτ 4/3 1, without tears! Also note that conjugating a kernel does not change its Fredholm determinant. Before proving Proposition.1, we shall need the following identities: Lemma. Introducing the polynomial Ψx, s; ω := 1 4 ω4 + 3 ω s 4sxω ω 3,.5 the Airy kernel 1.6 and the double integral part of the extended Airy kernel 1.5, at t 1 = s, t = s, satisfy the following differential equations, Ψx, s, x Ψy, s, y + 4s K A x, y = x yx + y + 6s K A x, y, and Ψx, s, x Ψy, s, y 3 s s x y K A x, y s, s = x yx + y + 6s K A x, y. s, s Proof: The operator on the left hand side of.6 reads Ψx, s, x Ψy, s, y + 4s = 4s1 + y y 3 y + x x 3 x + 3 s x y x 4 y..8 8

9 Using the first representation 1.6 of the Airy kernel, one checks using the differential equation for the Airy kernel, A x = xax and thus A x = xa x + Ax and A iv x = A x + x Ax, and using differentiation by parts to establish the last equality, y y y 3 + x x x 3 K A x, y = dz z ddz + Ax + zay + z = K A x, y. In order to take care of the other pieces in.8, one uses the second representation 1.6 of the kernel K A x, y, yielding x y K A x, y = x yk A x, y 4 x y 4 K A x, y = x y K A x, y. This establishes the first identity.6 of Lemma.. The operator on the left hand side of the second identity.7 reads: Ψx, s, x Ψy, s, y 3 s s x y = x 4 y + 3 s x y + 4sx x 3 x y y + 3 y 3 s s x y, of which we will evaluate all the different terms acting on the integral. Notice at first that, since the expression under the differentiation vanishes at and, one has = = s dz z ze sz K A x + z, y + z dz e sz K A x + z, y + z zdz e sz K A x + z, y + z, zdz e sz Ax + zay + z.9 Again, using the second representation 1.6 of the kernel K A x, y, one checks x y KA x, y = x y dz e sz K A x + z, y + z s, s x y KA x, y = x y K A x, y, s, s s, s 9

10 and also s x y KA s, s s x, y = sx y zdz e sz K A x + z, y + z and, using the differential equation x x x 3 Ax = Ax and.9, s x x x 3 y y y 3 KA x, y = sx y s, s zdz e sz K A x + z, y + z Using A iv x = A x+x Ax and the Darboux-Christoffel representation 1.6 of the Airy kernel, one checks 4 x y 4 KA x, y = s, s = x y e sz A iv x + zay + z Ax + za iv y + zdz +x y dz e sz K A x + z, y + z + x y K A x, y s, s zdz e sz Ax + zay + z. Adding all these different pieces and using the expression for sx y zdz e sz K A x + z, y + z, given by.9, leads to the statement of Lemma.. For the sake of notational convenience in the proof below, set K P τ 1,τ ξ, η dξ K11ξ P 1, η 1 dξ 1 dη 1 K1ξ P 1, η dξ 1 dη d η = K1ξ P, η 1 dξ dη 1 Kξ P, η.1 dξ dη and similarly for the Airy kernel K A t 1 t ; the K P ij and K A ij refer, as before, to the double integral part. Proof of Proposition.1: Notice that acting with x and y on the kernels K A x, y and K A x, y, amounts to multiplication of the integrand of the kernels with u and v respectively; i.e., u x and v y. 1

11 Given the small parameter z R, consider the following i, j-dependent change of integration variables U, V u, v in the four Pearcey kernels K P ij in.1, namely U = 1 3z uz t i z 4, V = 1 3z vz t j z 4,.11 together with the changes of variables ξ, η, τ i, τ j x, y, t i, t j, in accordance with. and.3, τ i = 1 3z t iz 4 + Oz 1, τ j = 1 3z t jz 4 + Oz 1 ξ = 7 3τ i 3/ 3τ i 1/6 x, η = 7 3τ j 3/ 3τ j 1/6 y..1 To be precise, for Kkk P, one sets in the transformations above i = j = k, for K1 P and K1, P one sets i = 1, j = and i =, j = 1 respectively. Then, remembering the expressions Φ and Ψ, defined in.1 and.5, the following estimate holds for small z: e Φy,t j;z 4 e V T j V V η e Φx,t i;z 4 e U4 4 + T i U Uξ Ψy,t j ;v = e z4 e v3 3 +yv e z4 Ψx,t i ;u e 1+t ioz 8 +t j Oz 8 u3 3 +xu Ψy,t j ; y = e z4 e v3 3 +yv e z4 Ψx,t i ; x e 1+t ioz 8 +t j Oz 8 ; u3 3 +xu.13 replacing in the latter expression u and v by differentiations x and y has the advantage that upon doubly integrating in u and v, the fraction of exponentials in front can be taken out of the integration. Moreover, setting t 1 = t + s and t = t s, one also checks 5 : e Φy,t j;z 4 e Φx,t i;z 4 pp τ j τ i ; ξ, η dξdη = p A s, x, y dxdy 1 + z4 [ x yx+y + 6s + t 4 s rx, y, s] + Oz The precise expression for rx, y, s := x y 8s x + y s will be irrelevant in the sequel. 11

12 The multiplication by the quotient of the exponentials on the left hand side of the expressions above will amount to a conjugation of the kernel, which will not change the Fredholm determinant. Setting t i = t + s and t j = t s, with t =, one finds for the non-exponential part in the kernel dudv dξdη V U dxdy dudv1 + 7 = t i + t j z 4 v + t j u t i + 3z 4 vt j ut i + Oz8 { dxdy dudv i = 1, j = 1 1 ± 4sz 4 + Oz 8 for v u i =, j = = dxdy dudv v u s 1 ± 3z4 su+v + Oz 8 v u s for { i = 1, j = i =, j = Along the same vein as the remark in the beginning of the proof of this Theorem, one notices that multiplication of the integrand of the kernel K A x, y with the fraction, appearing in the last formula of.15, amounts 1 1 to an appropriate differentiation, to wit: ± 3z4 su + v v u s 3 s s y x..16 So we have, using.13,.14,.15, and the differential equation for K A of Lemma., e Φy,t 1 ;z 4 e Φx,t 1 ;z 4 KP 11 ξ, η dξdη K A x, y dxdy t= = z 4 ±4s + Ψx, ±s, x Ψy, ±s, y K A x, y dxdy + Oz 8 = z4 4 x yx + y + 6s K A x, y dxdy + Oz 8 ;.17 1

13 the upperlower-indices correspond to the upperlower-signs. Using.16, and the differential equation for K A of Lemma., 1 1 e Φy,t ;z 4 1 K P ξ, η dξdη K A x, y dxdy e Φx,t 1 ;z t= 3 = z 4 s s y x + Ψx, ±s, x Ψy, s, y KA x, y dxdy = z4 + Oz 8 4 x yx + y + 6s K A x, y dxdy 1 + Oz 8. t= Also, using.14, 1 e Φy,t ;z 4 e pτ Φx,t 1;z 4 τ 1 ; ξ, η dξdη p A s, x, y dxdy = z4 4 x yx + y + 6s p A s, x, y dxdy + Oz t= t= Since hy, t i, z 4 hx, t j ; z 4 = z4 4 x yx+y+6s for arbitrary t i,j = t±s at t =, and thus e hy,t i;z 4 e hx,t j;z 4 = 1 + z4 4 x yx + y + 6s + Oz 8. Then the following approximations follow upon combining the three estimates above.17,.18,.19 and using.: e Φy,t 1 ;z 4 +hy,t 1 ;z 4 e Φx,t 1 ;z 4 +hx,t 1 ;z 4 KP 11 e Φy,t 1 ;z 4 +hy,t 1 ;z 4 e Φx,t 1 ;z 4 +hx,t 1 ;z 4 K P 1 1 ξ, η dξdη K A x, y dxdy = Oz 8 t= ξ, η dξdη K A x, y dxdy = Oz 8 1 t= e Φy,t ;z 4 +hy,t ;z 4 e Φx,t 1;z 4 +hx,t 1 ;z 4 pp τ τ 1 ; ξ, η dξdη p A s, x, y dxdy = Oz 8. t= The reader is reminded that this estimate sofar is done at t =. It then follows for t from the estimates.11,.1,.13,.14,.15, 13 1

14 .16 that the right hand side of.4 is an asymptotic series in z 4 with polynomial coefficients in t 1 + t = t, proving the claim about O 1. So far an important point was omitted, namely to analyze how the Pearcey contour turns in the limit into an Airy contour; a detailed description of the possible contours was given in Figure. The Pearcey-rays X 1, with angles σ 1, σ 1 π/8, 3π/8 can be deformed into two acceptable rays θ 1, θ 1 π/6, π/ for the Γ < -Airy contour, since the two intervals have a non-empty intersection. Also, since 3π/8, 5π/8 π/6, π/, the Pearcey Y -contour can be deformed into an acceptable v Γ > -Airy contour. Again, since the admissible interval π/8, 3π/8 for σ and σ in the X -Pearcey contour contains π/6, one ends up in the limit integrating the function e u3 /3 along a contour of the form Γ >, which we may choose to have an angle of π/6 δ with the negative real axis, for small arbitrary δ >. In the sector Γ >, with < θ = θ π/6 δ, the function e u3 /3 decays exponentially fast. Therefore, the contribution, due to the u-integration of e u3 /3 lower order terms over Γ >, having an angle of π/6 δ with the negative real axis, vanishes by applying Cauchy s Theorem in that sector. This ends the proof of Proposition.1. 3 From the Pearcey to the Airy statistics This section concerns itself with proving Theorem 1.1. Proof of Theorem 1.1: For any intervals E 1 and E, the function Qτ 1, τ ; E 1, E = log P Pτ i E i = i=1 3.1 satisfies the Pearcey PDE Reparametrizing, without loss of generality, τ 1 = τ +σ, τ = τ σ, E 1 = ξ+η+µ, ξ+η µ, leads to a manageable PDE for the function E = ξ η+ν, ξ η ν, 3. F τ, σ; ξ, η, µ, ν := Qτ 1, τ ; E 1, E,

15 namely the PDE: 3 F τ σ σ τ τ + ξ ξ + η η + µ µ + ν F σ τ ξ η + { F τ ξ, F ξ ν F ξ } ξ =. 3.4 To go from the Pearcey PDE 1.11 to the PDE 3.4, one notices that the operators 1.1, appearing in the Pearcey PDE 1.11, have simple expressions in terms of the variables τ, σ, ξ, η, µ, ν, τ Q = F τ, EQ = F ξ, ε τ Q := σ σ + τ τ F, ε EQ = ξ ξ + η η + µ µ + ν F, ν i τ i Ei Q = τ ξ + σ F. η The change of variables, considered in. and.3, combined with a linear change of variables, in parallel with 3., τ i = 1 { 3z 1 + 6t iz 4 t1 = t + s with 6 t = t s E i = { 7 3τ i 3/ 3τ i 1/6 Ẽ1 = x + y + u, x + y u Ẽ i with Ẽ = x y + v, x y v 3.5 yields a z-dependent invertible map, T : t, s, x, y, u, v τ, σ, ξ, η, µ, ν, 3.6 and thus the function F in 3.3 leads to a new z-dependent function G: F τ, σ; ξ, η, µ, ν = F T t, s, x, y, u, v =: Gt, s, x, y, u, v, of which we compute, in principle, the series in z. From Proposition.1, it follows that the z 4 -term in the asymptotic expansion in.4 vanishes when t = ; so, omitting dxdy, one has SK P τ 1,τ S 1 = K A t 1,t + z 4 K 1 + z 4i K i, with K 1 = th, 15

16 for some kernel H, with the K i polynomial in t. Then defining one finds: L i := I K A t 1,t 1 K i, 3.7 Gt, s, x, y, u, v = F τ, σ; ξ, η, µ, ν = log P {Pτ i E i = } i=1 3.8 = log deti K P τ 1,τ E1 E = log deti K A t 1,t Ẽ1 Ẽ z 4 Tr L 1 z 8 TrL + 1 L 1... =: G s; Ẽ1, Ẽ G 1 t, s; Ẽ1, Ẽz 4 G t, s; Ẽ1, Ẽz 8 + Oz 1, where G s; Ẽ1, Ẽ = log deti K A t 1,t Ẽ1 Ẽ, G 1 t, s; Ẽ1, Ẽ = t TrI K A t 1,t 1 HẼ1 Ẽ. 3.9 Note G s; Ẽ1, Ẽ is t-independent, since the Airy process is stationary. Then setting F τ, σ; ξ, η, µ, ν = GT 1 τ, σ; ξ, η, µ, ν = Gt, s, x, y, u, v into the PDE 3.4 yields, using differentiation by parts, a new PDE for Gt, s, x, y, u, v. Upon setting the expansion 3.8 of G for small z, Gt, s, x, y, u, v = G s; Ẽ1, Ẽ G 1 t, s; Ẽ1, Ẽz 4 + Oz 8, into this new PDE leads to z s 3 G s x 3 G 1 + Oz 6 =, 3.1 t x with G 1 being polynomial in t. From the term 3 F in the PDE 3.4, it τ 3 would seem like the leading term would be of order z 6 ; in fact there are two consecutive cancellations, so that the first non-trivial term has order 16

17 z, thus leading to a simple PDE connecting G 1 with G. But since G is t-independent, equation 3.1 can also be written as G t x 1 ts s G = 3.11 From 3.9, one finds G 1 ts s G = t TrI K A t 1,t 1 s HẼ1 Ẽ s G = l t i a i s; x, y, u, v, 1 which substituted back in 3.11, leads to the PDE s for the a i, namely implying a i s; x, y, u, v =, 3.1 x a i s; x, y, u, v = xb i s; y, u, v + c i s; y, u, v Letting the intervals Ẽ1 and Ẽ go to, while keeping their relative position y fixed and widths u and v fixed as well, is achieved by letting x, as follows from 3.. Remember t 1, t, Ẽ1, Ẽ from 3.5. But in the limit x, the expressions G, namely G s; Ẽ1, Ẽ = log deti K A t 1,t Ẽ1 Ẽ tends to exponentially fast using the exponential decay of the four components of the matrix Airy kernel 1.5; note that the term p A t, x, y tends to as well, when x and y tend to, due to the presence of e tx+y/. Next, we sketch the proof that G 1 t, s; Ẽ1, Ẽ = t TrI K A t 1,t 1 HẼ1 Ẽ tends to exponentially fast, when x. Indeed, using the identity R is the resolvent of the Airy kernel K A t 1,t I K A t 1,t 1 H = H + K A t 1,t I K A t 1,t 1 H = H + K A t 1,t I + RH, 17

18 one computes where G 1 = TrI K A t 1,t 1 K 1 Ẽ1 Ẽ = t TrI K A t 1,t 1 HẼ1 Ẽ = t + t TrK A TrHẼ1 Ẽ t 1,t, with H := I + RH, HẼ1 Ẽ = H TrHẼ1 Ẽ ii u, udu i=1 Ẽ i TrK A t 1,t = K HẼ1 Ẽ t A i t i u, v H ii v, ududv i=1 Ẽ i Ẽi + Kt A 1 t u, v H 1 v, ududv Ẽ 1 Ẽ + Kt A t 1 u, v H 1 v, ududv. Ẽ Ẽ1 Each of these integrals tend to because each of them contains the Airy kernel and also because H is obtained by acting with differential operators on the Airy kernel, as explained in section. This ends the proof that G 1 t, s; Ẽ1, Ẽ, when the Ẽi tend to. This fact together with the form 3.13 of the a i imply b i s; y, u, v = c i s; y, u, v = for i 1, and thus a i = for i 1, implying G 1 = ts s G. Summarizing, this implies that for E i = 3τ 7 i 3/ 3τ i 1/6 Ẽ i, substituting G 1 = ts G / s into 3.8, { Pτi 7 log P 3τ } i 3/ 3τ i 1/6 Ẽi = i=1 = G s; Ẽ1, Ẽ z 4 ts G s + Oz8 = G s tsz 4 ; Ẽ1, Ẽ + Oz 8 { = log P At i 1 t i z 4 Ẽi = } + Oz 8, i=1 18

19 in view of the change of variables given in 3.5, one has that s tsz 4 = 1 t 11 t 1 z 4 t 1 t z 4. Then, substituting t i = u i 1 + u i z 4 into t i 1 t i z 4 and τ i = 1 + 6t i z 4 /3z 6 + Oz 1, as in., yields t i 1 t i z 4 = u i u 3 i z 8 u 4 i z 1 and τ i = 1 3z 6 +u i z +u i z +Oz 1 for i = 1,. Then eliminating z between the two expressions above for τ 1 and τ, by first expressing z as a series in τ 1, yields 1.1 and 1., with t i replaced by u i. This ends the proof of Theorem 1.1. References [1] Mark Adler, M. Cafasso, and Pierre van Moerbeke. Integrable Kernels, Integrable Systems and PDEs forthcoming 1 [] Mark Adler, Nicolas Orantin, and Pierre van Moerbeke. Universality of the Pearcey process. Physica D, 39:94-941, 1. ArXiv:91.45 [3] Mark Adler and Pierre van Moerbeke. PDEs for the joint distributions of the Dyson, Airy and sine processes. Ann. Probab., 334: , 5. ArXiv:math.PR/339 and math.pr/4354 [4] Mark Adler and Pierre van Moerbeke. PDEs for the Gaussian ensemble with external source and the Pearcey distribution. Comm. Pure Appl. Math., 69:161 19, 7. ArXiv:math.PR/5947 [5] Alexander I. Aptekarev, Pavel M. Bleher, and Arno B. J. Kuijlaars. Large n limit of Gaussian random matrices with external source. II. Comm. Math. Phys., 59: , 5. ArXiv:math-ph/4841 [6] M. Bertola and M. Cafasso. The Riemann-Hilbert approach to the transition between the gap probabilities from the Pearcey to the Airy process. ArXiv:15.483, 1. [7] P. M. Bleher and A. B. J. Kuijlaars. Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not., 3:19 19, 4. ArXiv:math-ph/

20 [8] Pavel Bleher and Arno B. J. Kuijlaars. Large n limit of Gaussian random matrices with external source. I. Comm. Math. Phys., 51-3:43 76, 4. ArXiv:math-ph/44 [9] Pavel M. Bleher and Arno B. J. Kuijlaars. Large n limit of Gaussian random matrices with external source. III. Double scaling limit. Comm. Math. Phys., 7: , 7. ArXiv:math-ph/664 [1] Mark J. Bowick and Édouard Brézin. Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett. B, 681:1 8, [11] E. Brézin and S. Hikami. Correlations of nearby levels induced by a random potential. Nuclear Phys. B, 4793:697 76, ArXiv:condmat/96546 [1] E. Brézin and S. Hikami. Extension of level spacing universality. Phys Rev., E 56:64 69, ArXiv:cond-mat/9713 [13] E. Brézin and S. Hikami. Level spacing of random matrices in an external source. Phys. Rev. E 3, 586, part A: , ArXiv:condmat/9844 [14] E. Brézin and S. Hikami. Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E 3, 574: , ArXiv:cond-mat/9843 [15] Freeman J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Mathematical Phys., 3: , 196. [16] Kurt Johansson. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys., 153:683 75, 1. ArXiv:math-ph/6 [17] Kurt Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys., 41-:77 39, 3. ArXiv:math/68 [18] Kurt Johansson. The arctic circle boundary and the Airy process. Ann. Probab., 331:1 3, 5. ArXiv:math/3616

21 [19] M. L. Mehta. Random matrices, volume 14 of Pure and Applied Mathematics Amsterdam. Elsevier/Academic Press, Amsterdam, third edition, 4. [] Gregory Moore. Matrix models of D gravity and isomonodromic deformation. In Random surfaces and quantum gravity Cargèse, 199, volume 6 of NATO Adv. Sci. Inst. Ser. B Phys., pages Plenum, New York, [1] Andrei Okounkov and Nicolai Reshetikhin. Random skew plane partitions and the Pearcey process. Comm. Math. Phys., 693:571 69, 7. ArXiv:math/5358 [] Michael Prähofer and Herbert Spohn. Scale invariance of the PNG droplet and the Airy process. J. Statist. Phys., 185-6: ,. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. ArXiv:math/154 [3] Craig A. Tracy and Harold Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys., 1591: , ArXiv:hepth/ [4] Craig A. Tracy and Harold Widom. Differential equations for Dyson processes. Comm. Math. Phys., 51-3:7 41, 4. ArXiv:math/398 [5] Craig A. Tracy and Harold Widom. The Pearcey process. Comm. Math. Phys., 63:381 4, 6. ArXiv:math/415 [6] P. Zinn-Justin. Random Hermitian matrices in an external field. Nuclear Phys. B, 4973:75 73, ArXiv:cond-mat/97333 [7] P. Zinn-Justin. Universality of correlation functions of Hermitian random matrices in an external field. Comm. Math. Phys., 1943:631 65, ArXiv:cond-mat/