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arXiv · 1809.03448

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

Thomas Leblé
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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 $C^4$, and $\mathcal{C}$ is a random point configuration distributed according to Sine-$β$, the integral of $ϕ(\cdot / \ell)$ against the random fluctuation $d\mathcal{C} - dx$, converges in law, as $\ell$ 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.

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Thomas Leblé. 2018-09-10. CLT for fluctuations of linear statistics in the Sine-beta process. https://arxiv.org/abs/1809.03448

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