Nonlinear PDEs for Fredholm determinants arising from string equations

TL;DR

This paper explores the nonlinear PDEs satisfied by Fredholm determinants of integrable kernels linked to string equations, revealing their connections to Painlevé systems, random matrix theory, and statistical mechanics.

Contribution

It introduces a systematic approach to derive nonlinear PDEs for Fredholm determinants associated with kernels from string equations, highlighting their integrable structure and applications.

Findings

Fredholm determinants satisfy specific nonlinear PDEs.

Connections established between kernels, Painlevé systems, and string relations.

Applications demonstrated in random matrix theory and Ising models.

Abstract

String equations related to 2D gravity seem to provide, quite naturally and systematically, integrable kernels, in the sense of Its-Izergin-Korepin and Slavnov. Some of these kernels (besides the "classical" examples of Airy and Pearcey) have already appeared in random matrix theory and they have a natural Wronskian structure, given by one of the operators in the string relation $[L^\pm,Q^\pm] = \pm 1$, namely $L^\pm$. The kernels are intimately related to wave functions for Gel'fand-Dickey reductions of the KP hierarchy. The Fredholm determinants of these kernels also satisfy Virasoro constraints leading to PDEs for their log derivatives, and these PDEs depend explicitly on the solutions of Painlev\'e-like systems of ODEs equivalent to the relevant string relations. We give some examples coming from critical phenomena in random matrix theory (higher order Tracy-Widom distributions) and statistical mechanics (Ising models).