Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

TL;DR

This paper solves the loop equations of the hermitian 2-matrix model to all orders, expressing correlation functions as residues on an algebraic curve with diagrammatic rules, including a cubic-vertex representation.

Contribution

It introduces a residue formula for all topological orders of the 2-matrix model correlation functions with diagrammatic rules, advancing the understanding of matrix models.

Findings

All-order solutions to the 2-matrix model loop equations

Residue representations on algebraic curves

Diagrammatic rules including cubic-vertex graphs

Abstract

We solve the loop equations of the hermitian 2-matrix model to all orders in the topological $1/N^2$ expansion, i.e. we obtain all non-mixed correlation functions, in terms of residues on an algebraic curve. We give two representations of those residues as Feynman-like graphs, one of them involving only cubic vertices.