From sine kernel to poisson statistics

Abstract

We study the Sineβprocess introduced in [B. Valkó and B. Virág. Invent. math. 177 463-508 (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 Sineβpoint process converges weakly to a Poisson point process on R. Thus, the Sineβpoint processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to β = ∞) and the Poisson process.