Matrix Models for Random Partitions

TL;DR

This paper develops exact matrix integral representations for sums over partitions, highlighting the role of logarithmic and exponential potentials, and demonstrates their Toda lattice integrability.

Contribution

It introduces novel matrix models with specific potentials for sums over partitions, derived using higher Casimir operators, and connects them to Toda lattice integrability.

Findings

Derived exact matrix integral representations for sums over partitions.

Identified the presence of logarithmic and exponential terms in the potentials.

Established Toda lattice integrability of the models.

Abstract

We derive exact matrix integral representations for different sums over partitions. The characteristic feature of all obtained matrix models is the presence of logarithmic (or, vice versa, exponential) terms in the potential. Our derivation is based on the application of the higher Casimir operators. The Toda lattice integrability of the basic sums over partitions can be easily derived from the matrix model representation.