2002/11/13 by N. M. Ercolani, Ercolani, N. M., K. D. T-R McLaughlin +1
Mathematics · Physics and Astronomy · #82D30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:82D30
paper · pdf · doi:10.48550/arxiv.math-ph/0211022
44 pages, 4 figures. To appear, International Mathematics Research Notices
arxiv created 2002/11/13 · arxiv updated 2009/11/30
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.