Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The right tail exponent of the Tracy–Widom $\beta$ distribution

Laure Dumaz and Bálint Virág

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The Tracy–Widom $\beta$ distribution is the large dimensional limit of the top eigenvalue of $\beta$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a\to\infty$ the tail of the Tracy–Widom distribution satisfies

\[P(\mathit{TW}_{\beta}>a)=a^{-(3/4)\beta+\mathrm{o}(1)}\exp\biggl(-\frac{2}{3}\beta a^{3/2}\biggr).\]


La loi de Tracy–Widom $\beta$ est la limite de la plus grande valeur propre des ensembles $\beta$ de matrices aléatoires lorsque leur taille tend vers l’infini. Nous utilisons la représentation par l’opérateur stochastique d’Airy pour montrer que lorsque $a\to\infty$ la queue de la loi de Tracy–Widom vérifie :

\[P(\mathit{TW}_{\beta}>a)=a^{-(3/4)\beta+\mathrm{o}(1)}\exp\biggl(-\frac{2}{3}\beta a^{3/2}\biggr).\]

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 4 (2013), 915-933.

First available in Project Euclid: 2 October 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60H25: Random operators and equations [See also 47B80]

Tracy–Widom distribution Stochastic Airy operator Beta ensembles


Dumaz, Laure; Virág, Bálint. The right tail exponent of the Tracy–Widom $\beta$ distribution. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 4, 915--933. doi:10.1214/11-AIHP475.

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