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) τ-functions, which is actually a combination of these two complementary requirements.
Highlights
Integrability describes the partition function in terms of the Grassmannian, and the string equation picks up a concrete point in the Grassmannian
As a byproduct of recent studies of far more complicated problems in [1] and [2], we introduced a set of formulas, some of them being probably new, which do provide a full solution of the Hermitian and rectangular complex models
The partition functions have very explicit character expansions and arbitrary Gaussian correlators are represented by finite sums
Summary
As a byproduct of recent studies of far more complicated problems in [1] and [2], we introduced a set of formulas, some of them being probably new, which do provide a full solution of the Hermitian and rectangular complex models. The partition functions have very explicit character expansions and arbitrary Gaussian correlators are represented by finite sums.
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