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

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Álvarez Galindo, Gabriel and Martínez Alonso, Luis and Medina Reus, Elena (2011) An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model. Nuclear Physics B, 848 (2). pp. 398-429. ISSN 0550-3213

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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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© 2011 Elsevier B.V. The financial support of the Universidad Complutense under project GR58/08-910556 and the Comisión Interministerial de Ciencia y Tecnología under projects FIS2008-00200 and FIS2008-00209 are gratefully acknowledged.

Uncontrolled Keywords:Graphical Enumeration, Partition-Function, Asymptotics, Universality, Behavior, Gravity, Polynomials, Limit, Hermitian Matrix Model, Genus Expansion, Counting Maps
Subjects:Sciences > Physics > Physics-Mathematical models
ID Code:20467
Deposited On:21 Mar 2013 11:26
Last Modified:10 Dec 2018 15:09

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