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From Sine kernel to Poisson statistics

We study the Sine$_β$ process introduced in [B. Valkó and B. Virág. Invent. math. (2009)] when the inverse temperature $β$ tends to 0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of $β$-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$_β$ point process converges weakly to a Poisson point process on $\mathbb{R}$. Thus, the Sine$_β$ point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to $β=\infty$) and the Poisson process.