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

作者：Alvarez Gabriel^*; Martinez Alonso Luis; Medina Elena

来源：Nuclear Physics B, 2011, 848(2): 398-429.

DOI：10.1016/j.nuclphysb.2011.02.019

• Hermitian matrix model
• Genus expansion
• Counting maps

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.

• 出版日期2011-7-11