We consider complex sample covariance matrices
= (1/)* where is a × random matrix with i.i.d. entries
, 1 ≤ ≤ , 1 ≤ ≤ , with distribution . Under some regularity and decay assumptions on , we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit where → ∞ and lim→∞ / = for any real number ∈ (0, ∞).

We consider some random band matrices with band-width ${N}^{\mu}$ whose entries are independent random variables with distribution tail in ${x}^{-\alpha}$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $\alpha lt;2(1+{\mu}^{-1})$, the largest eigenvalues have order ${N}^{(1+\mu )/\alpha}$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov...

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar–Parisi–Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent = 3/2, that means one should find a universal space–time limiting process under the scaling of time as , space like 2/3 and fluctuations like 1/3 as → ∞. In this paper we provide...

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in (
(2004) 82–91), we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.

Download Results (CSV)