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This paper is concerned with the asymptotic behavior of the free energy for aclass of Hermitean random matrix models, with odd degree polynomial potential,in the large N limit. It continues an investigation initiated and developed ina sequence of prior works whose ultimate aim is to reveal and understand, in arigorous way, the deep connections between correlation functions foreigenvalues of these random matrix ensembles on the one hand and theenumerative interpretations of their matrix moments in terms of mapcombinatorics (a branch of graph theory) on the other. In doing this we makeessential use of the link between the asymptotics of the random matrixpartition function and orthogonal polynomials with exponential weight equal tothe random matrix potential. Along the way we develop and analyze the continuumlimits of both the hierarchy of Toda lattice equations and the hierarchy ofdifference string equations associated to these orthogonal polynomials. Theformer are found to have the structure of a hierarchy of near-conservation lawsuniversal in the potential; the latter are a novel semi-classical extension ofthe traditional string equations. Our methods apply to regular maps of botheven and odd valence, however we focus on the latter since that is the relevantcase for this paper. These methods enable us to rigorously determine closedform expressions for the generating functions that enumerate trivalent maps, ingeneral implicitly, but also explictly in a number of cases.

The Continuum Limit of Toda Lattices for Random Matrices with Odd Weights