A computation of the covariance between two linear statistics for the Jellium model

TL;DR

This paper derives an exact formula for the covariance between two linear statistics in the one-dimensional Jellium model, extending previous results on variance and using asymptotic approximations and saddle-point methods.

Contribution

It provides a novel exact covariance formula for linear statistics in the Jellium model, based on large deviation and saddle-point techniques.

Findings

Derived an explicit covariance formula for linear statistics in Jellium

Extended previous variance results to covariance between two functions

Utilized asymptotic probability distribution and saddle-point approximation

Abstract

We extend previous results providing an exact formula for the variance of a linear statistic for the Jellium model, a one-dimensional model of Statistical mechanics obtained from the $k \longrightarrow 0^{+}$ limit of the Dyson log-gas. For such a computation of the covariance, in comparison to previous work for computations of the log-gas covariance, we obtain a formula between two linear statistics, given arbitrary functions $f$ and $g$ over the real line, that is dependent upon an asymptotic approximation of the Jellium probability distribution function from large $N$ deviations, and from an effective saddle-point action.