Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

The continuum limit of Toda lattices for random matrices with odd weights

Pages: 267 – 305



Nicholas M. Ercolani (Department of Mathematics, The University of Arizona)

Virgil U. Pierce (Department of Mathematics, The University of Texas – Pan American)


This paper is concerned with the asymptotic behavior of the free energy for a class of Hermitian random matrix models, with odd degree polynomial potential, in the large N limit. It continues an investigation initiated and developed in a sequence of prior works whose ultimate aim is to reveal and understand, in a rigorous way, the deep connections between correlation functions for eigenvalues of these random matrix ensembles on the one hand and the enumerative interpretations of their matrix moments in terms of map combinatorics (a branch of graph theory) on the other. In doing this we make essential use of the link between the asymptotics of the random matrix partition function and orthogonal polynomials with exponential weight equal to the random matrix potential. Along the way we develop and analyze the continuum limits of both the Toda lattice equations and the difference string equations associated to these orthogonal polynomials. The former are found to have the structure of a hierarchy of near-conservation laws; the latter are a novel semi-classical extension of the traditional string equations. One has these equations for each class of regular maps of a given valence. Our methods apply to regular maps of both even and odd valence, however we focus on the latter since that is the relevant case for this paper. These methods enable us to rigorously determine closed form expressions for the generating functions that enumerate trivalent maps, in general implicitly, but also explicitly in a number of cases.


random matrices, Toda lattice, Motzkin paths, string equations, conservation law hierarchies, map enumeration

2010 Mathematics Subject Classification

05C30, 34M55, 60B20

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