DLR equations and rigidity for the Sine-beta process
vendredi 26 avril 2019, 9h30
Joint work with: Thomas Leblé, Adrien Hardy and Mylène Maïda,
We investigate properties of Sine-beta process, 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. We adopt a statistical physics perspective, and give a description of the Sine-beta process using the Dobrushin-Lanford-Ruelle (DLR) formalism: the restriction of Sine-beta process 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 the Sine-beta process 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.