Schur function expansion for normal matrix model and associated discrete matrix models

TL;DR

This paper explores the Schur function expansion of the normal matrix model's partition function, linking it to Toda lattice tau functions and discrete matrix models solvable via orthogonal polynomials.

Contribution

It establishes a connection between the Schur expansion of normal matrix models and Toda hierarchy tau functions, also deriving discrete versions of classical matrix models.

Findings

Schur function expansion matches Takasaki's Toda tau function expansion.

Partition function of normal matrices equals that of certain discrete models.

Discrete models include non-negative, unitary, and normal matrices.

Abstract

We consider Schur function expansion for the partition function of the model of normal matrices. We show that this expansion coincides with Takasaki expansion \cite{Tinit} for tau functions of 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.