Decay of correlations and thermodynamic limit for the circular Riesz gas

TL;DR

This paper proves that the microscopic configuration of a circular Riesz gas converges to an infinite volume measure as the number of particles grows, using correlation decay and elliptic regularity methods.

Contribution

It establishes the thermodynamic limit for the circular Riesz gas and analyzes correlation decay using Helffer-Sj"ostrand equations and discrete elliptic estimates.

Findings

Correlation decay with power-law exponent 2-s.

Convergence of particle configurations to an infinite volume measure.

Application of elliptic regularity to long-range interactions.

Abstract

We investigate the thermodynamic limit of the circular long-range Riesz gas, a system of particles interacting pairwise through an inverse power kernel. We show that after rescaling, so that the typical spacing of particles is of order $1$, the microscopic point process converges as the number of points tends to infinity, to an infinite volume measure $\mathrm{Riesz}_{s,\beta}$. This convergence result is obtained by analyzing gaps correlations, which are shown to decay in power-law with exponent $2-s$. Our method is based on the analysis of the Helffer-Sj\"ostrand equation in its static form and on various discrete elliptic regularity estimates.