Number-Rigidity and $\beta$-Circular Riesz gas

TL;DR

This paper proves the existence of a non number-rigid Gibbs point process called the $eta$-circular Riesz gas for certain non integrable potentials, marking a first in this area.

Contribution

It introduces the $eta$-circular Riesz gas and demonstrates its non number-rigidity, expanding understanding of Gibbs processes with non integrable interactions.

Findings

Existence of $eta$-circular Riesz gas for all $d \\ge 1$, $eta > 0$

First proof of non number-rigidity for a Gibbs process with non integrable potential

Uses a statistical physics approach based on canonical DLR equations

Abstract

For an inverse temperature $\beta>0$, we define the $\beta$-circular Riesz gas on $\mathbb{R}^d$ as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential $g(x) = \Vert x \Vert^{-s}$. We focus on the non integrable case $d-1<s<d$. Our main result ensures, for any dimension $d\ge 1$ and inverse temperature $\beta>0$, the existence of a $\beta$-circular Riesz gas which is not number-rigid. Recall that a point process is said number rigid if the number of points in a bounded Borel set $\Delta$ is a function of the point configuration outside $\Delta$. It is the first time that the non number-rigidity is proved for a Gibbs point process interacting via a non integrable potential. We follow a statistical physics approach based on the canonical DLR equations. It is inspired by Dereudre-Hardy-Lebl\'e and Ma\"ida (2021) where the authors prove the number-rigidity of the $\text{Sine}_\beta$ process.