Universal scaling limits of matrix models, and (p,q) Liouville gravity

TL;DR

This paper establishes universal scaling limits for matrix models near critical points, linking eigenvalue correlation functions to (p,q)-minimal models kernels, and provides explicit forms for these kernels in terms of solutions to linear differential equations.

Contribution

It introduces a universal framework connecting matrix model eigenvalue correlations to (p,q) minimal models, with explicit kernel constructions for the 1-matrix case.

Findings

Correlation functions near critical points are given by determinants of (p,q) kernels.

The (p,q) kernels are expressed via solutions to linear equations with polynomial coefficients.

Special cases recover known kernels like the Airy kernel for regular edges.

Abstract

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ~ x^{p/q}, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p,q)-minimal models kernels. Those (p,q) kernels are written in terms of functions solutions of a linear equation of order q, with polynomial coefficients of degree at most p. For example, near a regular edge y ~ x^{1/2}, the (1,2) kernel is the Airy kernel and we recover the Airy law. Those kernels are associated to the (p,q) minimal model, i.e. the (p,q) reduction of the KP hierarchy solution of the string equation. Here we consider only the 1-matrix model, for which q=2.