2017/08/11 by Tom Claeys, Claeys, Tom, Antoine Doeraene +1
Mathematics · Computer Science · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Bayesian Methods and Mixture Models
paper · pdf · doi:10.48550/arxiv.1708.03481
For a wide class of Hermitian random matrices, the limit distribution of the\neigenvalues close to the largest one is governed by the Airy point process. In\nsuch ensembles, the limit distribution of the k-th largest eigenvalue is given\nin terms of the Airy kernel Fredholm determinant or in terms of Tracy-Widom\nformulas involving solutions of the Painlev 'e II equation. Limit distributions\nfor quantities involving two or more near-extreme eigenvalues, such as the gap\nbetween the k-th and the \ℓ-th largest eigenvalue or the sum of the k largest\neigenvalues, can be expressed in terms of Fredholm determinants of an Airy\nkernel with several discontinuities. We establish simple Tracy-Widom type\nexpressions for these Fredholm determinants, which involve solutions to systems\nof coupled Painlev 'e II equations, and we investigate the asymptotic behavior\nof these solutions.\n