Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Overcrowding asymptotics for the $\operatorname{Sine}_{\beta}$ process

Diane Holcomb and Benedek Valkó

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Abstract

We give overcrowding estimates for the $\operatorname{Sine}_{\beta}$ process, the bulk point process limit of the Gaussian $\beta$-ensemble. We show that the probability of having exactly $n$ points in a fixed interval is given by $e^{-\frac{\beta}{2}n^{2}\log(n)+\mathcal{O}(n^{2})}$ as $n\to\infty$. We also identify the next order term in the exponent if the size of the interval goes to zero.

Résumé

Nous obtenons des résultats asymptotiques pour le surpeuplement du processus $\operatorname{Sine}_{\beta}$, le processus ponctuel limite dans le milieu du spectre de l’ensemble $\beta$-gaussien. Nous montrons que la probabilité d’observer $n$ points dans un interval fixé est donné par la formule $e^{-\frac{\beta}{2}n^{2}\log(n)+\mathcal{O}(n^{2})}$ quand $n\to\infty$. Nous obtenons aussi une approximation jusqu’à l’ordre suivant lorsque la longueur de l’interval tend vers $0$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1181-1195.

Dates
Received: 30 June 2015
Revised: 3 March 2016
Accepted: 10 March 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624035

Digital Object Identifier
doi:10.1214/16-AIHP752

Mathematical Reviews number (MathSciNet)
MR3689965

Zentralblatt MATH identifier
1381.60021

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F10: Large deviations 15B52: Random matrices

Keywords
$\beta$-ensembles Random matrices Overcrowding

Citation

Holcomb, Diane; Valkó, Benedek. Overcrowding asymptotics for the $\operatorname{Sine}_{\beta}$ process. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1181--1195. doi:10.1214/16-AIHP752. https://projecteuclid.org/euclid.aihp/1500624035


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