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Second order asymptotics for matrix models. (English) Zbl 1129.15020

The asymptotical behaviour of Hermitian random matrices whose distribution is given by a small convex perturbation of the Gaussian unitary ensemble is studied. \(m\)-tuples of random matrices are considered. The first order correction to the free energy is expressed as a generating function for the enumeration of maps. This is done by proving a central limit theorem for some specific polynomials and also for arbitrary polynomials. An interpretation of the variance and of the free energy in terms of a generating function for the number of planar maps is given. Finally, the second order correction to the free energy is computed.

MSC:

15B52 Random matrices (algebraic aspects)
05C30 Enumeration in graph theory
60F05 Central limit and other weak theorems
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