An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model
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.
Cite
@article{arxiv.1101.2727,
title = {An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model},
author = {Gabriel Álvarez and Luis Martínez Alonso and Elena Medina},
journal= {arXiv preprint arXiv:1101.2727},
year = {2011}
}