Selberg Integrals, Super hypergeometric functions and Applications to $\beta$-Ensembles of Random Matrices

TL;DR

This paper introduces a new Selberg-type integral related to deformed Calogero-Sutherland systems, explores its properties, and applies it to evaluate characteristic polynomial ratios in beta-ensembles of Random Matrix Theory.

Contribution

It presents a novel Selberg-type integral, establishes its differential equations, and connects a hypergeometric function with super Jack polynomials to Random Matrix Theory applications.

Findings

Derived a holonomic system for the integral.

Proved uniqueness of the hypergeometric function solution.

Evaluated characteristic polynomial ratios in beta-ensembles.

Abstract

We study a new Selberg-type integral with $n+m$ indeterminates, which turns out to be related to the deformed Calogero-Sutherland systems. We show that the integral satisfies a holonomic system of $n+m$ non-symmetric linear partial differential equations. We also prove that a particular hypergeometric function defined in terms of super Jack polynomials is the unique solution of the system. Some properties such as duality relations, integral formulas, Pfaff-Euler and Kummer transformations are also established. As a direct application, we evaluate the expectation value of ratios of characteristic polynomials in the classical $\beta$-ensembles of Random Matrix Theory.