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

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 alpha > 0. Using a Coulomb gas method, we study the statistics of s = (1/N) sigma(N)(i=1) f(x(i)) 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 Var(s) asymptotic to b/N-3 for large N. In this paper, we provide an exact compact formula for the con-stant b = 1/(4 alpha) integral(2 alpha)(-2 alpha) [f '(x)](2) dx. In addition, we also calculate the full large deviation function characterizing the tails of the full distribution P(s,N) for several different examples of f (x). Our analytical predictions are confirmed by numerical simulations.