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.