The asymptotic expansion of the partition function of random matrices

Abstract

We explore two methods for calculating the Taylor Coefficients of the terms of the asymptotic expansion of the partition function of random matrices for specific even potentials. The first of these methods applies to the leading order term. We show that this term has an elementary form in terms of a solution to an algebraic equation. This generates a general formula for the Taylor Coefficients of this term. Next we exploit the relationship between orthogonal polynomials and the Toda Lattice Equations to derive ODE's for the general terms of the expansion of the partition function of random matrices, which leads to a method of calculating the Taylor Coefficients of these functions.