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Asymptotics of the partition function of a random matrix model. (English) Zbl 1135.82016
Summary: 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.

##### MSC:
 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 15B52 Random matrices (algebraic aspects) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
##### Keywords:
Matrix Models; orthogonal polynomials; partition function
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