Publication : t10/137
Large deviations of the maximal eigenvalue of random matrices
Borot G. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)Eynard B. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)
Majumdar S.N. (Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), Université Paris-Sud XI, F-91405 Orsay, FRANCE)
Nadal C. (Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), Université Paris-Sud XI, F-91405 Orsay, FRANCE)
Abstract:Année de publication : 2011
We present detailed computations of the 'at least finite' terms (three dominant orders) of the free energy in a one-cut matrix model with a hard edge a, in beta-ensembles, with any polynomial potential. beta is a positive number, so not restricted to the standard values beta = 1 (hermitian matrices), beta = 1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This model allows to study the statistic of the maximum eigenvalue of random matrices. We compute the large deviation function to the left of the expected maximum. We specialize our results to the gaussian beta-ensembles and check them numerically. Our method is based on general results and procedures already developed in the literature to solve the Pastur equations (also called "loop equations"). It allows to compute the left tail of the analog of Tracy-Widom laws for any beta, including the constant term.
Revue : J. Stat. Mech. 2011 P11024 (2011)
DOI : 10.1088/1742-5468/2011/11/P11024
Preprint : arXiv:1009.1945
Keywords : Extreme value statistics, Tracy-Widom laws, beta ensembles, large deviations, double scaling limit
Langue : Anglais
