Gegenbauer polynomials and fluctuation properties of the one-dimensional Riesz gas

TL;DR

This paper analyzes the fluctuation properties of the one-dimensional Riesz gas using Gegenbauer polynomials, providing explicit covariance formulas for linear statistics in the infinite density limit.

Contribution

It introduces a Gegenbauer polynomial basis approach to compute covariance of linear statistics for the Riesz gas, generalizing previous cosine-based methods.

Findings

Derived covariance formulas in terms of Gegenbauer polynomials.

Reduced the power sum statistic covariance to a gamma function product.

Validated formulas against recent exact results in the literature.

Abstract

The Riesz gas in one-dimension consists of particles interacting via a pair potential, ${\rm sgn}(s) |x - x'|^{-s}$, $s \ne 0$ and $-\log | x - x'|$ for $s=0$. In the infinite density limit, with the particle support the interval $[-1,1]$, we apply a functional derivative method due to Beenakker to compute the covariance of two smooth linear statistics for the Riesz gas with exponent $s \in (-1,1)$, $s \ne 0$. This we give in terms of a sum over Fourier components of the linear statistics with respect to a Gegenbauer polynomial $\{C_n^{(s/2)}(x) \}$ basis, which generalises a known form in the case $s=0$ involving a cosine expansion. For the power sum linear statistic, our general formula can be reduced to a product of gamma function form, and compared against recent exact results in the literature for this case.