Singularity dominated strong fluctuations for some random matrix averages

TL;DR

This paper rigorously analyzes the asymptotic behavior of averaged characteristic polynomial moduli in circular and Jacobi random matrix ensembles, confirming a conjecture and revealing singularity-dominated fluctuations near eigenvalue support.

Contribution

It establishes the leading asymptotic form of these averages using hypergeometric functions based on Jack polynomials, confirming a conjecture of Berry and Keating.

Findings

Averages diverge for certain parameter ranges near eigenvalue support

The asymptotic form is expressed via generalized hypergeometric functions

Confirmed a conjecture related to singularity behavior in circular ensembles

Abstract

The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval $(0,1)$ of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power $2 \mu$ diverges, for $2\mu \le -1$, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.