Abstract
AbstractWe investigate Sineβ, 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 Sineβ using the Dobrushin‐Lanford‐Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sineβ 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, Sineβ is a natural infinite Gibbs measure at inverse temperature β > 0 associated with the logarithmic pair potential interaction. Moreover, we show that Sineβ 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.
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