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Number-rigidity and \(\beta\)-circular Riesz gas. (English) Zbl 07690055

Summary: 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 \(g(x)=\| x \|^{-s}\). We focus on the nonintegrable case \(d-<s<d\). Our main result ensures, for any dimension \(d \geq 1\) 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 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 [D. Dereudre et al., Commun. Pure Appl. Math. 74, No. 1, 172–222 (2021; Zbl 1467.82013)] where the authors prove the number-rigidity of the \(\mathrm{Sine}_{\beta}\) process.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics

Citations:

Zbl 1467.82013
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References:

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