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Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices

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Abstract

In this paper we derive analytic characterizations for and explicit evaluations of the coefficients of the matrix integral genus expansion. The expansion itself arises from the large N asymptotic expansion of the logarithm of the partition function of N × N Hermitian random matrices. Its g th coefficient is a generating function for graphical enumeration on Riemann surfaces of genus g. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree 2ν. Our results are based on a hierarchy of recursively solvable differential equations, derived through a novel continuum limit, whose solutions are the coefficients we want to characterize. These equations are interesting in their own right in that their form is related to partitions of 2g + 1 and joint probability distributions for conditioned random walks.

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Authors and Affiliations

  1. Dept. of Math., Univ. of Arizona, Tucson, AZ, 85721, USA

    N. M. Ercolani & K. D. T-R. McLaughlin

  2. Dept. of Math., The Ohio State University, Columbus, OH, 43210, USA

    V. U. Pierce

Authors
  1. N. M. Ercolani
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  2. K. D. T-R. McLaughlin
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  3. V. U. Pierce
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Correspondence to N. M. Ercolani.

Additional information

Communicated by M. Aizenman

K. D. T-R McLaughlin was supported in part by NSF grants DMS-0451495 and DMS-0200749, as well as a NATO Collaborative Linkage Grant “Orthogonal Polynomials: Theory, Applications, and Generalizations” Ref no. PST.CLG.979738.

N. M. Ercolani and V. U. Pierce were supported in part by NSF grants DMS-0073087 and DMS-0412310.

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Ercolani, N.M., McLaughlin, K.D.TR. & Pierce, V.U. Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices. Commun. Math. Phys. 278, 31–81 (2008). https://doi.org/10.1007/s00220-007-0395-z

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