An exact formula for the variance of linear statistics in the one-dimensional jellium model

Abstract

We consider the jellium model of N particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength α > 0. Using a Coulomb gas method, we study the statistics of where f(x), in principle, is an arbitrary smooth function. While the mean of s is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance for large N. In this paper, we provide an exact compact formula for the constant . In addition, we also calculate the full large deviation function characterizing the tails of the full distribution for several different examples of f(x). Our analytical predictions are confirmed by numerical simulations.