Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlev\'e II equation

TL;DR

This paper demonstrates that certain multiplicative eigenvalue statistics of Hermitian random matrices converge to universal limits described by an integro-differential Painlevé II equation, linking random matrix theory with KPZ universality.

Contribution

It establishes universality of multiplicative eigenvalue statistics for Hermitian matrices and connects these limits to the integro-differential Painlevé II equation.

Findings

Universal limits described by integro-differential Painlevé II equation

Connection between random matrix eigenvalues and KPZ equation

Convergence holds for a large class of potentials and statistics

Abstract

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix limit several associated quantities converge to limits which are universal in both the potential and the family of multiplicative statistics considered. In turn, such universal limits are described by the integro-differential Painlev\'e II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the KPZ equation at any finite time.