From sine kernel to Poisson statistics

Abstract

We study the Sine $\beta$ process introduced in Valko and Virag, 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 characterised 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 the real line. 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.