An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

TL;DR

This paper introduces an efficient method to compute genus expansions of free energy in Hermitian matrix models, utilizing orthogonal polynomials, Bleher-Its deformation, and Toda hierarchy resolvent, enabling enumeration of labeled maps.

Contribution

It presents a novel approach combining orthogonal polynomial recurrence, Bleher-Its deformation, and Toda hierarchy techniques to compute genus expansions and map enumeration efficiently.

Findings

Developed an algorithm for genus expansion computation from recurrence coefficients.

Provided a new method for counting labeled k-maps without explicit coefficients.

Addressed regularization of singular one-cut models within the framework.

Abstract

We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher-Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.