  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 $\tau_tV(z)$ of $V(z)$ to $z^2$, $0\le
t<\infty$ 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 $N^{-2}$ 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 $N^{-2}$ of the free energy, for a $V$, which
admits a one-cut regular deformation $\tau_tV$; (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 $V$.
