From the Pearcey to the Airy Process.

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. IntroductionPutting 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 [2] governing the gap probabilities for the Pearcey process. The Airy process was introduced by Prähofer-Spohn [22] 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 [23,24]. The Pearcey process was introduced in [25,21] in the context of non-intersecting Brownian motions and plane partitions, also based on prior work on matrix models with external source [20,

The Gap Probabilities of the Tacnode, Pearcey and Airy Point Processes, Their Mutual Relationship and Evaluation

“…• In the second section, as a sort of warm-up, we show how to use the theoretical results in [6] to compute (numerically) the gap probability of the Pearcey process. As an application, we give numerical confirmations of the degeneration of the Pearcey gap probability to a couple of Tracy-Widom distribution (see [6] and, for a similar result, [1]).…”

“…Here we will not study the rate of convergence in formulas (3.28), (3.29). We simply point out that, if we knew some equation for the tacnode gap probability, we could perform a similar analysis to the one in [1]. The extension to multi-time case (and multi-intervals), though more cumbersome, does not present any additional difficulty.…”

Edge fluctuations of limit shapes

“…The full arrows represent cases that have been proved, [1], [12] or that would be relatively straightforward to prove, whereas the dotted arrows are not known. At the top of the hierarchy one might expect a sufficiently general form of the Discrete Tacnode kernel.…”

Transitions Between Critical Kernels: From the Tacnode Kernel and Critical Kernel in the Two-Matrix Model to the Pearcey Kernel

“…Physically, this will lead to descriptions of phase transitions among different processes. For similar transitions between canonical processes, we refer to the thesis of Deschout [25] and the recent papers [2,6,7].…”