Asymptotics of the partition function of a random matrix model

Abstract

We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of N of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free energy for a singular quartic potential. We also prove the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of the potential, and the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular potential.