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

Walid Hachem (1) , Adrien Hardy (2) , Jamal Najim (3)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 LIGM - Laboratoire d'Informatique Gaspard-Monge

The aim of this note is to provide a pedagogical survey of the recent works by the authors ( arXiv:1409.7548 and arXiv:1507.06013) 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.

Mots clés: Large random matrices; Wishart matrices; local fluctuations; Airy kernel; Bessel kernel; Pearcey kernel

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]; Statistiques [math.ST]

Walid Hachem, Adrien Hardy, Jamal Najim. A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices. ESAIM: Proceedings and Surveys, 2015, 51, pp.150-174. ⟨hal-01199904⟩