From Sine kernel to Poisson statistics

We study the Sine$_\beta$ process introduced in [B. Valk\'o and B. Vir\'ag. Invent. math. (2009)] when the inverse temperature $\beta$ tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of $\beta$-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sine$_\beta$ point process converges weakly to a Poisson point process on $\mathbb{R}$. Thus, the Sine$_\beta$ point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $\beta=\infty$) and the Poisson process.