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.

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Cited by in corpus (4)

• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• Determination of S-curves with applications to the theory of nonhermitian orthogonal polynomials
• Partition functions and the continuum limit in Penner matrix models
• Solution of the string equations for asymmetric potentials