CLT for fluctuations of linear statistics in the Sine-beta process

2018/09/10 by Leblé, Thomas · 2 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1809.03448

Abstract

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 C⁴, and C is a random point configuration distributed according to Sine-β, the integral of ϕ(⋅ / ℓ) against the random fluctuation dC - dx, converges in law, as ℓ 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.

Cited by

• Central limit theorem for the sine-β point process at β≤ 2
• On the pair correlation function of the Sine_β process

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