Nonlinear PDEs for gap probabilities in random matrices and KP theory

TL;DR

This paper establishes a deep connection between kernels in random matrix theory, their Fredholm determinants, and nonlinear PDEs derived from KP hierarchy reductions, providing a systematic approach to analyze gap probabilities.

Contribution

It demonstrates that kernels related to random matrix theory are connected to KP hierarchy wave functions and their Fredholm determinants satisfy linear and nonlinear PDEs, advancing theoretical understanding.

Findings

Fredholm determinants satisfy linear PDEs (Virasoro constraints)

Nonlinear PDEs for gap probabilities are derived systematically

Connections between kernels, KP hierarchy, and random matrix processes are established

Abstract

Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are intimately related to wave functions for polynomial (Gel'fand-Dickey reductions) or rational reductions of the KP-hierarchy; their Fredholm determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a systematic way, non-linear PDEs for the Fredholm determinant of such kernels. Examples include Fredholm determinants giving the gap probability of some infinite-dimensional diffusions, like the Airy process, with or without outliers, and the Pearcey process, with or without inliers.