From the Pearcey to the Airy process

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.

Consider n nonintersecting Brownian particles on the real line R, −∞ < x 1 (t) < ... < x n (t) < ∞ with (local) Brownian transition probability given by all starting from the origin x = 0 at time t = 0 and such that n 2 particles are forced to ± n 2 at t = 1. For very large n, the average mean density of particles has its support, for t ≤ 1 2 , on one interval centered about x = 0, and for 1 2 < t ≤ 1 on two intervals symmetrically located about x = 0. The end points of the interval(s) of support describe a heart-shaped region in (x, t)-space, with a cusp at (x 0 , t 0 ) = (0, 1/2). The Pearcey process P(τ ) is defined (see Figure 1) as the motion of these nonintersecting Brownian motions for large n, about (x 0 , t 0 ) = (0, 1/2) (i.e., near the cusp), with space microscopically rescaled by a factor of n −1/4 and time rescaled by a factor n −1/2 , in tune with the Brownian motion rescaling. A partial differential equation for the Pearcey process was found by Adler-van Moerbeke [4] and in [2] 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 = 2 27 (3(t − t 0 )) 3/2 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: Theorem 1.1 Given finite parameters t 1 < t 2 , let both τ 1 , τ 2 → ∞, such that τ 2 − τ 1 → ∞ behaves in the following precise way: The parameters t 1 and t 2 provide the Airy times in the following approximation of the Airy process by the Pearcey process: 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 2.1, 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 [2] and Proposition 2.1, 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 , of the matrix kernel in x = (x 1 , . . . , x ) and y = (y 1 , . . . , y ), denoted as 4 χ E is the indicator function for the set E 3 follows: 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]) 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 2 and θ 2 with the negative real axis, as indicated in Figure 2. As is well known, one may choose θ 1 , θ 2 , θ 1 , θ 2 = π/3. In particular, for t i = t j , this is the customary Airy kernel Pearcey contour e e e π/6 < θ 1 , θ 2 , θ 1 , θ 2 < π/2 Airy contour  The Pearcey process is determinantal as well, for which the multi-time gap probability is given by the Fredholm determinant for τ 1 < . . . < τ , for the Pearcey matrix kernel in ξ = (ξ 1 , . . . , ξ ) and η = (η 1 , . . . , η ), denoted as follows: where for arbitrary τ i and τ j , (see Tracy-Widom [25]) where X and Y are the following contours: X = X 1 ∪ X 2 consists of four rays emanating from the origin with angles σ 1 , σ 1 with the positive real axis and σ 2 , σ 2 with the negative real axis, as given in Figure 2. 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 2 = σ 1 = σ 2 = π/4 and τ = τ = π/2. In [4,2], it was shown that, given intervals the log of the probability satisfies a non-linear PDE, which we describe here. Given the times and the intervals above, one defines the following operators: (1.10) Then Q satisfies the Pearcey partial differential equation in its arguments and the boundary of the intervals:

From the Pearcey to the Airy kernel
Define the rational functions and the diagonal matrix We now state: 1 Given a parameter z → 0, define two times τ 1 and τ 2 blowing up like z −6 and depending on two parameters t 1 and t 2 , Given large ξ i and η j , define new space variables x i and y j , using τ i above, 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 2 , where O 1 refers to a differential operator in ∂/∂x, ∂/∂y, with polynomial coefficients in x, y, t 1 ± t 2 and (t 1 − t 2 ) −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 (1.2) ), without tears! Also note that conjugating a kernel does not change its Fredholm determinant. Before proving Proposition 2.1, we shall need the following identities: the Airy kernel (1.6) and the (double) integral part of the extended Airy kernel (1.5), at t 1 = s, t 2 = −s, satisfy the following differential equations,

6)
and (2.7) Proof: The operator on the left hand side of (2.6) reads (2.8) Using the first representation (1.6) of the Airy kernel, one checks using the differential equation for the Airy kernel, A (x) = xA(x) and thus A (x) = xA (x) + A(x) and A (iv) (x) = 2A (x) + x 2 A(x), and using differentiation by parts to establish the last equality, In order to take care of the other pieces in (2.8), one uses the second representation (1.6) of the kernel K A (x, y), yielding This establishes the first identity (2.6) of Lemma 2.2. The operator on the left hand side of the second identity (2.7) reads: 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 0 and ∞, one has Again, using the second representation (1.6) of the kernel K A (x, y), one checks and also and, using the differential equation Using A (iv) (x) = 2A (x)+x 2 A(x) and the Darboux-Christoffel representation (1.6) of the Airy kernel, one checks Adding all these different pieces and using the expression for −2s(x − y) ∞ 0 zdz e −2sz K A (x + z, y + z), given by (2.9), leads to the statement of Lemma 2.2.
For the sake of notational convenience in the proof below, set and similarly for the Airy kernel K A t 1 t 2 ; theK P ij andK A ij refer, as before, to the (double) integral part.
Proof of Proposition 2.1: Notice that acting with ∂ x and ∂ y on the kernels K A 11 22 (x, y) and K A 12 21 (x, y), amounts to multiplication of the integrand of the kernels with −u and v respectively; i.e., u ↔ −∂ x and v ↔ ∂ y .
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 (2.10), namely together with the changes of variables (ξ, η, τ i , τ j ) → (x, y, t i , t j ), in accordance with (2.2) and (2.3), (2.12) To be precise, for K P kk , one sets in the transformations above i = j = k, for K P 12 and K P 21 , one sets i = 1, j = 2 and i = 2, j = 1 respectively. Then, remembering the expressions Φ and Ψ, defined in (2.1) and (2.5), the following estimate holds for small z: (2.13) 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 2 = t − s, one also checks 5 : (2.14) 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 = 0, one finds for the non-exponential part in the kernel 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 12 21 (x, y) with the fraction, appearing in the last formula of (2.15), amounts to an appropriate differentiation, to wit: (2.18) Also, using (2.14), (

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 2 , the function satisfies the Pearcey PDE (1.11). Reparametrizing, without loss of generality, 2) leads to a manageable PDE for the function F (τ, σ; ξ, η, µ, ν) := Q(τ 1 , τ 2 ; E 1 , E 2 ), (3.3) 14 namely the PDE: To go from the Pearcey PDE (1.11) to the PDE (3.4), one notices that the operators (1.10), appearing in the Pearcey PDE (1.11), have simple expressions in terms of the variables τ, σ, ξ, η, µ, ν, The change of variables, considered in (2.2) and (2.3), combined with a linear change of variables, in parallel with (3.2), 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)) =: G(t, s, x, y, u, v), of which we compute, in principle, the series in z. From Proposition 2.1, it follows that the z 4 -term in the asymptotic expansion in (2.4) vanishes when t = 0; so, omitting √ dxdy, one has for some kernel H, with the K i polynomial in t. Then defining one finds: Note G 0 (s;Ẽ 1 ,Ẽ 2 ) is t-independent, since the Airy process is stationary. Then setting F (τ, σ; ξ, η, µ, ν) = G(T −1 (τ, σ; ξ, η, µ, ν)) = G(t, s, x, y, u, v) into the PDE (3.4) yields, using differentiation by parts, a new PDE for G(t, s, x, y, u, v). Upon setting the expansion (3.8) of G for small z, into this new PDE leads to with G 1 being polynomial in t. From the term ∂ 3 F ∂τ 3 in the PDE (3.4), it 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 z 2 , thus leading to a simple PDE connecting G 1 with G 0 . But since G 0 is t-independent, equation (3.10) can also be written as ∂ ∂t From (3.9), one finds Letting the intervalsẼ 1 andẼ 2 go to ∞, while keeping their relative position 2y fixed and widths −2u and −2v fixed as well, is achieved by letting x → ∞, as follows from (3.2). Remember t 1 , t 2 ,Ẽ 1 ,Ẽ 2 from (3.5). But in the limit x → ∞, the expressions G 0 , namely G 0 (s;Ẽ 1 ,Ẽ 2 ) = log det(I − K A t 1 ,t 2 )Ẽ 1 ×Ẽ 2 tends to 0 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 0 as well, when x and y tend to ∞, due to the presence of e −t(x+y)/2 . Next, we sketch the proof that G 1 (t, s;Ẽ 1 ,Ẽ 2 ) = t Tr((I − K A t 1 ,t 2 ) −1 H)˜E one computes Each of these integrals tend to 0 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 2. This ends the proof that G 1 (t, s;Ẽ 1 ,Ẽ 2 ) → 0, when theẼ i 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) = 0 for i ≥ 1, and thus a i = 0 for i ≥ 1, implying Summarizing, this implies that for E i = 2 27 (3τ i ) 3/2 − (3τ i ) 1/6Ẽ i , substituting G 1 = 2ts(∂G 0 /∂s) into (3.8),