Publication: An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model
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2011-07-11
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Elsevier Science BV
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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.
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