Schur function expansions of matrix integrals

A. Yu. Orlov

Abstract: We consider the Schur function expansion of various matrix integrals.
(Based partly on joint work with J.Harnad, T. Shiota and further developments.)
(1) We consider two-matrix models with an interaction term that generalizes the standard one, analyzed by Itsykson and Zuber.
(2) We also consider Schur function expansion for the partition function of the model of normal matrices. We show that this expansion coincides with the Takasaki expansion for tau functions of the Toda lattice hierarchy. We show that the partition function of the model of normal matrices is, at the same time, a partition function of certain discrete models, which can be solved by the method of orthogonal polynomials. We obtain discrete versions of various known matrix models: models of non-negative matrices, unitary matrices, normal matrices.
(The latter is based on a joint paper with T.Shiota)