DLR equations and rigidity for the Sine-beta process

Abstract : 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-Landford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine β 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. 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.
Complete list of metadatas

Contributor : Adrien Hardy <>
Submitted on : Thursday, December 13, 2018 - 3:56:24 PM
Last modification on : Friday, April 19, 2019 - 4:54:52 PM
Long-term archiving on : Thursday, March 14, 2019 - 3:44:18 PM


Files produced by the author(s)


  • HAL Id : hal-01954367, version 1



David Dereudre, Adrien Hardy, Thomas Leblé, Mylène Maïda. DLR equations and rigidity for the Sine-beta process. 2018. ⟨hal-01954367⟩



Record views


Files downloads