Phase transition of the largest eigenvalue for non-null complex sample covariance matrices

Abstract

We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say $r$, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished $r$ eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of the random matrix theory. Especially a phase transition phenomena is observed. Our results also apply to a last passage percolation model and a queuing model.

50 pages, 8 figures

References in corpus (5)

• Level-Spacing Distributions and the Airy Kernel
• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Instabilities in complex mixtures with a large number of components
• Collective Origin of the Coexistence of Apparent RMT Noise and Factors in Large Sample Correlation Matrices
• Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles

Cited by in corpus (3)

• Generalizations of TASEP in discrete and continuous inhomogeneous space
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Dyson's non-intersecting Brownian motions with a few outliers