CLT for Fluctuations of Linear Statistics in the Sine-beta Process

We prove, for any β >0, a central limit theorem for the f luctuations of linear statistics in the Sineβ 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 C4, and C is a random point configuration distributed according to Sineβ, 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 H1/2 norm of ϕ on the real line. The proof relies on the Dobrushin-Landford-Ruelle equations for Sineβ 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.