An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model |
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Authors: | Gabriel Álvarez Luis Martínez Alonso Elena Medina |
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Institution: | 1. Departamento de Física Teórica II, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain;2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, 11510 Puerto Real, Cádiz, Spain |
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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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Keywords: | 14N10 82B41 15B52 |
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