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

For an inverse temperature β > 0, we define the β-circular Riesz gas on R 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 non integrable 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 ∆ is a function of the point configuration outside ∆. It is the first time that the non numberrigidity 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 the recent paper [4] where the authors prove the number-rigidity of the Sineβ process. key words: Gibbs point process, DLR equations, equivalence of ensembles. Introduction The pairwise Riesz potential g(x) = ‖x‖−s for x ∈ Rd is abundantly studied in several domains of mathematics as statistical mechanics, potential theory, optimization, etc. The particular case s = d− 2 corresponds to the the Coulomb pair potential coming from the electrostatic theory. The general case s ≤ d is particularly interesting and challenging since the potential is not integrable at infinity. The thermodynamic limits of associated canonical Gibbs measures at inverse temperature β > 0 are the natural microscopic equilibrium states appearing in the bulk of systems with a large number of particles. Their studies are old topics in physics and mathematical physics literature [12, 20]. Recently a general large deviation principle at the microscopic level has been established [18] with the rate function equals to the entropy plus the renormalized energy times β. The general description of thermodynamic limits or minimizers of 1 david.dereudre@univ-lille.fr, Univ. Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé, F-59000 lille, France. 2 thibaut.vasseur@univ-lille.fr, Univ. Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé, F-59000 lille, France.