DLR equations and rigidity for the Sine-beta process

David Dereudre (1) , Adrien Hardy (1, 2) , Thomas Leblé (3, 4) , Mylène Maïda (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Paradyse - Systèmes de particules et systèmes dynamiques
3 CIMS - Courant Institute of Mathematical Sciences [New York]
4 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We investigate Sine β , the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one-dimensional log-gases, or β-ensembles, at inverse temperature β > 0. We adopt a statistical physics perspective, and give a description of Sine β using the Dobrushin-Landford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine β to a compact set, conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration. In short, Sine β is a natural infinite Gibbs measure at inverse temperature β > 0 associated with the logarithmic pair potential interaction. Moreover, we show that Sine β is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough to include more general long range interactions in arbitrary dimension.

Domaines: Mathématiques [math]; Physique mathématique [math-ph]; Probabilités [math.PR]

David Dereudre, Adrien Hardy, Thomas Leblé, Mylène Maïda. DLR equations and rigidity for the Sine-beta process. Communications on Pure and Applied Mathematics, 2020, ⟨10.1002/cpa.21963⟩. ⟨hal-01954367⟩