Double scaling limit for matrix models with non analytic potentials

TL;DR

This paper investigates the double scaling limit in random matrix models with non-analytic potentials, deriving asymptotic expansions and connecting them to Painleve II and Dirac systems.

Contribution

It introduces a novel approach using perturbation expansion for string equations to analyze non-analytic potentials in matrix models.

Findings

Asymptotic expansion of Jacobi matrix entries obtained

First order terms linked to Painleve II solutions

Reproducing kernel expressed via Dirac system solutions

Abstract

We study the double scaling limit for unitary invariant ensembles of random matrices with non analytic potentials and find the asymptotic expansion for the entries of the corresponding Jacobi matrix. Our approach is based on the perturbation expansion for the string equations. The first order perturbation terms of the Jacobi matrix coefficients are expressed through the Hastings-McLeod solution of the Painleve II equation. The limiting reproducing kernel is expressed in terms of solutions of the Dirac system of differential equations with a potential defined by the first order terms of the expansion.