Universality of local eigenvalue statistics for some sample covariance matrices

Abstract

We consider random, complex sample covariance matrices 1 N X * X , where X is a p × N random matrix with i.i.d. entries of distribution µ. It has been conjectured that both the distribution of the distance between nearest neighbor eigenvalues in the bulk and that of the smallest eigenvalues become, in the limit N → ∞, p N → 1, the same as that identified for a complex Gaussian distribution µ. We prove these conjectures for a certain class of probability distributions µ.

Characteristic polynomials of sample covariance matrices: The non-square case

“…These results are in consistence with results by Ben Arous and Péché [4], Tao and Vu [24], Soshnikov [21], Péché [20], and Feldheim and Sodin [11], who obtained similar results for the (more relevant) correlation function of the eigenvalues, yet under stronger assumptions on the underlying distributions. Ben Arous and Péché [4] proved the occurrence of the sine kernel in the bulk of the spectrum for a certain class of complex sample covariance matrices with γ ∞ = 1. Very recently, Tao and Vu [24] extended this result to a quite general class of sample covariance matrices, still with γ ∞ = 1.…”

An Introduction to Random Matrices

“…Concerning the spacing in the bulk, universality was proved when the i.i.d. entries are complex and have a distribution that can be written as convolution with a Gaussian law, see [Joh01b] (for the complex Wigner case) and [BeP05] (for the complex Wishart case). The proof is based on an application of the Itzykson-Zuber-Harish-Chandra formula, see the bibliographical notes for Section 4.3.…”

Bulk universality for Wigner matrices

“…We fix a small positive constant c 1 1 and define the domains WD f´D x C iy W jx a N j "; jy b N j Ä c 1 "=2g; Recall that q N D q C N D a N C ib N from (3.24).…”