Asymptotics of the partition function of a random matrix model

We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential V (z). Our approach is based on a deformation τtV (z) of V (z) to z , 0 ≤ t < ∞ and on the use of the underlying integrable structures of the matrix model. The main results include (1) the existence of a full asymptotic expansion in powers of N of the recurrence coefficients of the related orthogonal polynomials, for a one-cut regular V ; (2) the existence of a full asymptotic expansion in powers of N of the free energy, for a V , which admits a one-cut regular deformation τtV ; (3) the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of V ; (4) the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular V ; (5) the double scaling asymptotics of the free energy for a singular quartic polynomial V .