Master loop equations, free energy and correlations for the chain of matrices

TL;DR

This paper derives explicit loop equations for a chain of hermitian matrices, computes key quantities like free energy and resolvents, and begins analyzing 1/N^2 corrections, linking algebraic geometry to matrix models.

Contribution

It provides explicit formulas for loop equations and related functions for hermitian matrix chains, including initial steps towards 1/N^2 expansion analysis.

Findings

Explicit loop equations including 1/N^2 corrections

Solution expressed via algebraic curve geometry

Computed free energy and resolvents

Abstract

The loop equations for a chain of hermitian random matrices are computed explicitely, including the 1/N^2 corrections. To leading order, the master loop equation reduces to an algebraic equation, whose solution can be written in terms of geometric properties of the underlying algebraic curve. In particular we compute the free energy, the resolvents, the 2-loop functions and some mixed one loop functions. We also initiate the calculation of the 1/N^2 expansion.