The right tail exponent of the Tracy–Widom distribution
Annales de l'I.H.P. Probabilités et statistiques (2013)
- Volume: 49, Issue: 4, page 915-933
- ISSN: 0246-0203
Access Full Article
top
Abstract
topHow to cite
topDumaz, Laure, and Virág, Bálint. "The right tail exponent of the Tracy–Widom $\beta $ distribution." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 915-933. <http://eudml.org/doc/272040>.
@article{Dumaz2013,
abstract = {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\rightarrow \infty $ the tail of the Tracy–Widom distribution satisfies \[P(\mathit \{TW\}\_\{\beta \}>a)=a^\{-(3/4)\beta +\mathrm \{o\}(1)\}\exp \biggl (-\frac\{2\}\{3\}\beta a^\{3/2\}\biggr ).\]},
author = {Dumaz, Laure, Virág, Bálint},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Tracy–Widom distribution; stochastic airy operator; beta ensembles; Tracy-Widom distribution; stochastic Airy operator},
language = {eng},
number = {4},
pages = {915-933},
publisher = {Gauthier-Villars},
title = {The right tail exponent of the Tracy–Widom $\beta $ distribution},
url = {http://eudml.org/doc/272040},
volume = {49},
year = {2013},
}
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
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 915
EP - 933
AB - 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\rightarrow \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 - eng
KW - Tracy–Widom distribution; stochastic airy operator; beta ensembles; Tracy-Widom distribution; stochastic Airy operator
UR - http://eudml.org/doc/272040
ER -
References
top- [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. MR2869955
- [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. Zbl06225664
- [4] Y. Chen and S. M. Manning. Some eigenvalue distribution functions of the Laguerre ensemble. J. Phys. A 29 (23) (1996) 7561–7579. Zbl0906.60093MR1425838
- [5] I. Dumitriu and A. Edelman. Matrix models for beta ensembles. J. Math. Phys. 43 (11) (2002) 5830–5847. Zbl1060.82020MR1936554
- [6] F. J. Dyson. Statistical theory of the energy levels of complex systems. II. J. Math. Phys.3 (1962) 157–165. Zbl0105.41604MR143557
- [7] P. J. Forrester. Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34. Princeton Univ. Press, Princeton, NJ, 2010. Zbl1217.82003MR2641363
- [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. Zbl1239.60005MR2813333
- [9] C. A. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1) (1994) 151–174. Zbl0789.35152MR1257246
- [10] C. A. Tracy and H. Widom. On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177 (3) (1996) 727–754. Zbl0851.60101MR1385083
- [11] B. Valkó and B. Virág. Large gaps between random eigenvalues. Ann. Probab. 38 (3) (2010) 1263–1279. Zbl1223.60009MR2674999
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.