DLR Equations and Rigidity for the Sine‐Beta Process

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‐Lanford‐Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine β to a compact set, conditionally on 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. © 2020 Wiley Periodicals, Inc.