Electronic Journal of Probability

From sine kernel to Poisson statistics

Romain Allez and Laure Dumaz

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 114, 25 pp.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60G17 60B20 60J60: Diffusion processes [See also 58J65]

Random matrices Diffusions Poisson point process Exit time problem

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Allez, Romain; Dumaz, Laure. From sine kernel to Poisson statistics. Electron. J. Probab. 19 (2014), paper no. 114, 25 pp. doi:10.1214/EJP.v19-3742. https://projecteuclid.org/euclid.ejp/1465065756

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