Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies

TL;DR

This paper introduces a method using Whitham hierarchies to analyze phase transitions in large N matrix models, linking free energy to tau-functions and describing critical phenomena as third-order transitions.

Contribution

It develops a novel approach to study phase transitions in matrix models via matched solutions of Whitham hierarchies, connecting spectral endpoints to differential equations.

Findings

Free energy is the quasiclassical tau-function of the hierarchy.

Third-order phase transitions occur when the number of cuts changes by one.

The Bleher-Eynard model illustrates merging and birth of cuts.

Abstract

We present a method to study phase transitions in the large N limit of matrix models using matched solutions of Whitham hierarchies. The endpoints of the eigenvalue spectrum as functions of the temperature are characterized both as solutions of hodograph equations and as solutions of a system of ordinary differential equations. In particular we show that the free energy of the matrix model is the quasiclassical tau-function of the associated hierarchy, and that critical processes in which the number of cuts changes in one unit are third-order phase transitions described by C1 matched solutions of Whitham hierarchies. The method is illustrated with the Bleher-Eynard model for the merging of two cuts. We show that this model involves also a birth of a cut.