A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices

¹ CNRS LTCI; Télécom ParisTech, 46 rue Barrault, 75046 Paris Cedex 13, France.
Email: walid.hachem@telecom-paristech.fr
² Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 10044 Stockholm, Sweden.
Email: ahardy@kth.se
³ CNRS LIGM; Université Paris-Est, Cité Descartes, 5 Boulevard Descartes, Champs sur Marne, 77 454 Marne la Vallée Cedex 2, France.
Email: najim@univ-mlv.fr

Abstract

The aim of this note is to provide a pedagogical survey of the recent works [30, 31] concerning the local behavior of the eigenvalues of large complex correlated Wishart matrices at the edges and cusp points of the spectrum: Under quite general conditions, the eigenvalues fluctuations at a soft edge of the limiting spectrum, at the hard edge when it is present, or at a cusp point, are respectively described by mean of the Airy kernel, the Bessel kernel, or the Pearcey kernel. Moreover, the eigenvalues fluctuations at several soft edges are asymptotically independent. In particular, the asymptotic fluctuations of the matrix condition number can be described. Finally, the next order term of the hard edge asymptotics is provided.