Number-rigidity and β-circular Riesz gas

Abstract

For an inverse temperature β>0, we define the β-circular Riesz gas on Rd as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential g(x)=‖x‖−s. We focus on the nonintegrable case d−1<s<d. Our main result ensures, for any dimension d≥1 and inverse temperature β>0, the existence of a β-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 Δ is a function of the point configuration outside Δ. It is the first time that the nonnumber-rigidity is proved for a Gibbs point process interacting via a nonintegrable potential. We follow a statistical physics approach based on the canonical DLR equations. It is inspired by the recent paper (Comm. Pure Appl. Math. 74 (2021) 172–222) where the authors prove the number-rigidity of the Sineβ process.