Averages of ratios of characteristic polynomials in circular beta-ensembles and super-Jack polynomials

TL;DR

This paper investigates the averages of ratios of characteristic polynomials in circular beta-ensembles, employing Jack polynomial theory to derive new expressions and dual relations for different beta values.

Contribution

It introduces three novel formulas for ratio averages using super-Jack polynomials and hyperdeterminants, advancing the theoretical understanding of circular beta-ensembles.

Findings

Derived three expressions for ratio averages

Established dual relations between different beta values

Connected ratio averages with super-Jack polynomials

Abstract

We study the averages of ratios of characteristic polynomials over circular $\beta$-ensembles, where $\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as sums of super-Jack polynomials and another one is given by a hyperdeterminant. As applications, we give dual relations for ratio averages between $\beta$ and $4/\beta$.