From sine kernel to Poisson statistics

Romain Allez (Weierstrass Institute)
Laure Dumaz (University of Cambridge)


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.

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Pages: 1-25

Publication Date: December 12, 2014

DOI: 10.1214/EJP.v19-3742


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