The right tail exponent of the Tracy-Widom β distribution
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 915-933.

La loi de Tracy-Widom β est la limite de la plus grande valeur propre des ensembles β 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→∞ la queue de la loi de Tracy-Widom vérifie :

P(𝑇𝑊 β >a)=a -(3/4)β+o(1) exp- 2 3 β a 3/2 .

The Tracy-Widom β distribution is the large dimensional limit of the top eigenvalue of β random matrix ensembles. We use the stochastic Airy operator representation to show that as a→∞ the tail of the Tracy-Widom distribution satisfies

P(𝑇𝑊 β >a)=a -(3/4)β+o(1) exp- 2 3 β a 3/2 .

DOI : 10.1214/11-AIHP475
Classification : 60F10, 60H25
Mots-clés : Tracy-Widom distribution, stochastic airy operator, beta ensembles 
@article{AIHPB_2013__49_4_915_0,
     author = {Dumaz, Laure and Vir\'ag, B\'alint},
     title = {The right tail exponent of the {Tracy-Widom} $\beta $ distribution},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {915--933},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {4},
     year = {2013},
     doi = {10.1214/11-AIHP475},
     mrnumber = {3127907},
     zbl = {1278.60012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP475/}
}
TY  - JOUR
AU  - Dumaz, Laure
AU  - Virág, Bálint
TI  - The right tail exponent of the Tracy-Widom $\beta $ distribution
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2013
SP  - 915
EP  - 933
VL  - 49
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/11-AIHP475/
DO  - 10.1214/11-AIHP475
LA  - en
ID  - AIHPB_2013__49_4_915_0
ER  - 
%0 Journal Article
%A Dumaz, Laure
%A Virág, Bálint
%T The right tail exponent of the Tracy-Widom $\beta $ distribution
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 915-933
%V 49
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/11-AIHP475/
%R 10.1214/11-AIHP475
%G en
%F AIHPB_2013__49_4_915_0
Dumaz, Laure; Virág, Bálint. The right tail exponent of the Tracy-Widom $\beta $ distribution. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 915-933. doi : 10.1214/11-AIHP475. http://www.numdam.org/articles/10.1214/11-AIHP475/

[1] J. Baik. Asymptotics of Tracy-Widom distribution functions. Preprint, 2006. Available at: http://www.cirm.univ-mrs.fr/videos/2006/exposes/28/Baik.pdf.

[2] G. Borot, B. Eynard, S. N. Majumdar and C. Nadal. Large deviations of the maximal eigenvalue of random matrices. ArXiv e-prints, September 2010. | MR

[3] L. O. Chekhov, B. Eynard and O. Marchal. Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach. ArXiv e-prints, September 2010. | MR | Zbl

[4] Y. Chen and S. M. Manning. Some eigenvalue distribution functions of the Laguerre ensemble. J. Phys. A 29 (23) (1996) 7561-7579. | MR | Zbl

[5] I. Dumitriu and A. Edelman. Matrix models for beta ensembles. J. Math. Phys. 43 (11) (2002) 5830-5847. | MR | Zbl

[6] F. J. Dyson. Statistical theory of the energy levels of complex systems. II. J. Math. Phys. 3 (1962) 157-165. | MR | Zbl

[7] P. J. Forrester. Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34. Princeton Univ. Press, Princeton, NJ, 2010. | MR | Zbl

[8] J. A. Ramírez, B. Rider and B. Virág. Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Amer. Math. Soc. 24 (2011) 919-944. | MR | Zbl

[9] C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1) (1994) 151-174. | MR | Zbl

[10] C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 (3) (1996) 727-754. | MR

[11] B. Valkó and B. Virág. Large gaps between random eigenvalues. Ann. Probab. 38 (3) (2010) 1263-1279. | MR

Cité par Sources :