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We prove a number of results concerning the large 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
of the recurrence coefficients of the related orthogonal polynomials for a one-cut
regular potential and the double scaling asymptotics of the free...
We give integral representations for multiple Hermite and multiple Laguerre polynomials
of both type I and II. We also show how these are connected with double integral
representations of certain kernels from random matrix theory.
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