Open Access
2014 From sine kernel to Poisson statistics
Romain Allez, Laure Dumaz
Author Affiliations +
Electron. J. Probab. 19: 1-25 (2014). DOI: 10.1214/EJP.v19-3742


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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Romain Allez. Laure Dumaz. "From sine kernel to Poisson statistics." Electron. J. Probab. 19 1 - 25, 2014.


Accepted: 12 December 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1334.60078
MathSciNet: MR3296530
Digital Object Identifier: 10.1214/EJP.v19-3742

Primary: 60G55
Secondary: 60B20 , 60G17 , 60J60

Keywords: Diffusions , Exit time problem , Poisson point process , random matrices

Vol.19 • 2014
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