CLT FOR FLUCTUATIONS OF LINEAR STATISTICS IN THE SINE-BETA PROCESS

Thomas Leblé (1)

1 CIMS - Courant Institute of Mathematical Sciences [New York]

We prove, for any β > 0, a central limit theorem for the fluctuations of linear statistics in the Sine β process, which is the infinite volume limit of the random microscopic behavior in the bulk of one-dimensional log-gases at inverse temperature β. If ϕ is a compactly supported test function of class C 4 , and C is a random point configuration distributed according to Sine β , the integral of ϕ(•/) against the random fluctuation dC − dx, converges in law, as goes to infinity, to a centered normal random variable whose standard deviation is proportional to the Sobolev H 1/2 norm of ϕ on the real line. The proof relies on the DLR equations for Sine β established by Dereudre-Hardy-Maïda and the author, the Laplace transform trick introduced by Johansson, and a transportation method previously used for β-ensembles at macroscopic scale.

Domaines: Analyse fonctionnelle [math.FA]; Physique mathématique [math-ph]; Probabilités [math.PR]

Thomas Leblé. CLT FOR FLUCTUATIONS OF LINEAR STATISTICS IN THE SINE-BETA PROCESS. 2020. ⟨hal-03040300⟩