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

Abstract: We prove, for any $\beta >0$, a central limit theorem for the fluctuations of linear statistics in the Sine-$\beta$ process, which is the infinite volume limit of the random microscopic behavior in the bulk of one-dimensional log-gases at inverse temperature $\beta$. If $\phi$ is a compactly supported test function of class $C^4$, and $\mathcal{C}$ is a random point configuration distributed according to Sine-$\beta$, the integral of $\phi(\cdot / \ell)$ 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 $\phi$ on the real line. The proof relies on the DLR equations for Sine-$\beta$ established by Dereudre-Hardy-Ma\"ida and the author, the Laplace transform trick introduced by Johansson, and a transportation method previously used for $\beta$-ensembles at macroscopic scale.