Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices |
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Authors: | N M Ercolani K D T-R McLaughlin V U Pierce |
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Institution: | (1) Dept. of Math., Univ. of Arizona, Tucson, AZ 85721, USA;(2) Dept. of Math., The Ohio State University, Columbus, OH 43210, USA |
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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.
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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