Hypergeometric Functions of Random Matrices and Quasimodular Forms

TL;DR

This paper explores the use of hypergeometric functions of random matrices as holomorphic observables in the Circular Unitary Ensemble, revealing a novel link to quasimodular forms and number theory.

Contribution

It introduces a new application of hypergeometric functions in random matrix theory, connecting them to quasimodular forms and high-dimensional asymptotics.

Findings

Expected derivatives have asymptotic expansions in terms of quasimodular forms

Establishes a new connection between CUE and number theory

Provides insights into high-dimensional behavior of random matrix functions

Abstract

Hypergeometric functions of complex matrices were introduced by James in multivariate statistics. These special functions play many roles in random matrix theory. The main goal of this paper is to suggest a new use for them as holomorphic observables of the Circular Unitary Ensemble. We analyze the high-dimensional behavior of the expected derivatives of these random analytic functions, and show that they admit asymptotic expansions which can be described in terms of quasimodular forms, giving an apparently new connection between the CUE and number theory.