Pair correlation function of the one-dimensional Riesz gas

Physical Review Research, vol. 5

Abstract

A method from random-matrix theory is used to calculate the pair correlation function of a one-dimensional gas of $N\ensuremath{\gg}1$ classical particles with a power-law repulsive interaction potential $u(x)\ensuremath{\propto}{|x|}^{\ensuremath{-}s}$ (a so-called Riesz gas). An integral formula for the covariance of single-particle operators is obtained, which generalizes known results in the limits $s\ensuremath{\rightarrow}\ensuremath{-}1$ (Coulomb gas) and $s\ensuremath{\rightarrow}0$ (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.

Authors 1

1. C. W. J. Beenakker corresponding Aachen Instituut-Lorentz

   Leiden University

   Affiliation as printed

   Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, Netherlands