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

Abstract

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.

29 pages; 7 figures; to be published in the "Proceedings of the Journ{é}es MAS 2014"

References in corpus (7)

• The Pearcey Process
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices
• Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• On the principal components of sample covariance matrices
• PDEs satisfied by extreme eigenvalues distributions of GUE and LUE
• Universality in Complex Wishart ensembles: The 2 cut case

Cited by in corpus (5)

• Cleaning large correlation matrices: tools from random matrix theory
• Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• Sharp detection in PCA under correlations: all eigenvalues matter
• Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case