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We investigate Sine$_\beta$, the universal point process arising as thethermodynamic limit of the microscopic scale behavior in the bulk ofone-dimensional log-gases, or $\beta$-ensembles, at inverse temperature$\beta>0$. We adopt a statistical physics perspective, and give a descriptionof Sine$_\beta$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by provingthat it satisfies the DLR equations: the restriction of Sine$_\beta$ to acompact set, conditionally to the exterior configuration, reads as a Gibbsmeasure given by a finite log-gas in a potential generated by the exteriorconfiguration. Moreover, we show that Sine$_\beta$ is number-rigid and tolerantin the sense of Ghosh-Peres, i.e. the number, but not the position, ofparticles lying inside a compact set is a deterministic function of theexterior configuration. Our proof of the rigidity differs from the usualstrategy and is robust enough to include more general long range interactionsin arbitrary dimension.

DLR equations and rigidity for the Sine-beta process