Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration

Details Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration

2002-11-13

arxiv.org/abs/math-ph/0211022v1

Physics

Mathematical Physics

44 pages, 4 figures. To appear, International Mathematics Research Notices

Scientific paper

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.

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