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DLR Equations and Rigidity for the Sine‐Beta Process

DLR Equations and Rigidity for the Sine‐Beta Process We 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications on Pure & Applied Mathematics Wiley

DLR Equations and Rigidity for the Sine‐Beta Process

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References (61)

Publisher
Wiley
Copyright
© 2021 Wiley Periodicals LLC
ISSN
0010-3640
eISSN
1097-0312
DOI
10.1002/cpa.21963
Publisher site
See Article on Publisher Site

Abstract

We 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.

Journal

Communications on Pure & Applied Mathematics – Wiley

Published: Jan 1, 2021

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