Statistical correlations in a Coulomb gas with a test charge

TL;DR

This paper confirms a conjectured expectation value in a Coulomb gas with a test charge and extends the analysis to higher cumulants using techniques from random-matrix theory.

Contribution

It validates a previous surmise about Coulomb gas correlations and generalizes it to higher cumulants through orthogonal polynomial methods.

Findings

Confirmed the surmised expectation value in Coulomb gas

Extended the analysis to higher cumulants

Utilized orthogonal polynomials on the unit circle

Abstract

A recent paper [Jokela et al, arxiv:0806.1491 9 June 2008] contains a surmise about an expectation value in a Coulomb gas which interacts with an additional charge xi that sits at a fixed position. Here I demonstrate the validity of the surmised expression and extend it to a certain class of higher cumulants. The calculation is based on the analogy to statistical averages in the circular unitary ensemble of random-matrix theory and exploits properties of orthogonal polynomials on the unit circle.