Matrix Models vs. Matrix Integrals

TL;DR

This paper reviews the connections between solutions of loop equations in Hermitean one-matrix models and specific matrix integrals, highlighting their properties and relation to Seiberg-Witten-Whitham theory.

Contribution

It clarifies the relationship between general solutions of matrix model loop equations and particular multi-cut solutions described by advanced mathematical theories.

Findings

Relation between loop equation solutions and matrix integrals

Description of multi-cut solutions using Seiberg-Witten-Whitham theory

Detailed analysis of a simple integral example

Abstract

In a brief review, we discuss interrelations between arbitrary solutions of the loop equations that describe Hermitean one-matrix model and particular (multi-cut) solutions that describe concrete matrix integrals. These latter ones enjoy a series of specific properties and, in particular, are described in terms of Seiberg-Witten-Whitham theory. The simplest example of ordinary integral is considered in detail.