Phase transitions and equilibrium measures in random matrix models

TL;DR

This paper investigates phase transitions in Hermitian random matrix models with polynomial potentials by analyzing equilibrium measures and their evolution, revealing new insights into the local behavior at phase transitions.

Contribution

It introduces a dynamical system approach to study equilibrium measures and systematically derives new results on phase transition behaviors in random matrix models.

Findings

Characterization of phase transition types and behaviors

Development of a dynamical system for equilibrium measures

New results on local behavior at phase transitions

Abstract

The paper is devoted to a study of phase transitions in the Hermitian random matrix models with a polynomial potential. In an alternative equivalent language, we study families of equilibrium measures on the real line in a polynomial external field. The total mass of the measure is considered as the main parameter, which may be interpreted also either as temperature or time. Our main tools are differentiation formulas with respect to the parameters of the problem, and a representation of the equilibrium potential in terms of a hyperelliptic integral. Using this combination we introduce and investigate a dynamical system (system of ODE's) describing the evolution of families of equilibrium measures. On this basis we are able to systematically derive a number of new results on phase transitions, such as the local behavior of the system at all kinds of phase transitions, as well as to review a number of known ones.