DLR equations and rigidity for the Sine-beta process

TL;DR

This paper characterizes the Sine-beta process as a Gibbs measure satisfying DLR equations and proves it exhibits number-rigidity, with a novel approach applicable to broader long-range interactions.

Contribution

It provides the first rigorous DLR formalism description of Sine-beta and establishes number-rigidity using a new, robust proof method adaptable to other long-range interaction models.

Findings

Sine-beta satisfies the DLR equations as a Gibbs measure.

Sine-beta exhibits number-rigidity, with particle count inside a set determined by the exterior.

The proof method is robust and applicable to general long-range interactions.

Abstract

We investigate Sine$_\beta$, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one-dimensional log-gases, or $\beta$-ensembles, at inverse temperature $\beta>0$. We adopt a statistical physics perspective, and give a description of Sine$_\beta$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine$_\beta$ 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. Moreover, we show that Sine$_\beta$ 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.