# On the complete perturbative solution of one-matrix models

## TL;DR

This paper discusses the complete solvability of Hermitian and rectangular complex matrix models, highlighting their simple character expansions, correlation calculations, and unique integrability properties as hypergeometric tau-functions.

## Contribution

It provides a comprehensive summary of the recent advances in the exact solvability of certain matrix models, emphasizing their special mathematical structure and relation to representation theory.

## Key findings

- Partition functions have simple character expansions.
- Correlators are finite sums over Young diagrams.
- Complete solvability is a unique property of hypergeometric tau-functions.

## 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.