On the complete perturbative solution of one-matrix models

Abstract

We summarize the recent results about complete solvability of Hermitian and rectangular complex matrix models. Partition functions have very simple character expansions with coefficients made from dimensions of representation of the linear group $GL(N)$, and arbitrary correlators in the Gaussian phase are given by finite sums over Young diagrams of a given size, which involve also the well known characters of symmetric group. The previously known integrability and Virasoro constraints are simple corollaries, but no vice versa: complete solvability is a peculiar property of the matrix model (hypergeometric) $\tau$-functions, which is actually a combination of these two complementary requirements.