Pair correlation function of the one-dimensional Riesz gas

TL;DR

This paper uses random-matrix theory to derive the pair correlation function for a one-dimensional Riesz gas with power-law interactions, providing a unified approach that generalizes known cases and reveals universal properties.

Contribution

It introduces an integral formula for the covariance of single-particle operators in the Riesz gas, extending results to a broad range of interaction exponents s.

Findings

Derived the pair correlation function for the Riesz gas.

Unified the Coulomb and log-gas cases within a single framework.

Found a universal large-N limit for the variance of the center of mass.

Abstract

A method from random-matrix theory is used to calculate the pair correlation function of a one-dimensional gas of $N\gg 1$ classical particles with a power law repulsive interaction potential $u(x)\propto |x|^{-s}$ (a socalled Riesz gas). An integral formula for the covariance of single-particle operators is obtained which generalizes known results in the limits $s\rightarrow -1$ (Coulomb gas) and $s\rightarrow 0$ (log-gas). As an application, we calculate the variance of the center of mass of the Riesz gas, which has a universal large-$N$ limit that does not depend on the shape of the confining potential.