Random matrix ensembles with an effective extensive external charge

TL;DR

This paper investigates special random matrix ensembles with an external charge effect, analyzing their properties using Jack polynomials and demonstrating universality in eigenvalue behavior.

Contribution

It introduces and analyzes ensembles with an external charge effect, proving properties with Jack polynomial theory and establishing universality classes for eigenvalues.

Findings

Reproducing property of the Poisson kernel proved using Jack polynomials.

Variance of linear statistics matches Dyson circular ensemble.

Global density calculated exactly for all even $eta$ values.

Abstract

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $\beta$, while for $\beta = 2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.