The Annals of Probability

Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

Walid Hachem, Adrien Hardy, and Jamal Najim

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Abstract

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy–Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin (hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale $N^{2}$.

Article information

Source
Ann. Probab. Volume 44, Number 3 (2016), 2264-2348.

Dates
Received: September 2014
Revised: February 2015
First available in Project Euclid: 16 May 2016

Permanent link to this document
http://projecteuclid.org/euclid.aop/1463410043

Digital Object Identifier
doi:10.1214/15-AOP1022

Mathematical Reviews number (MathSciNet)
MR3502605

Zentralblatt MATH identifier
1346.15035

Subjects
Primary: 15A52
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems

Keywords
Large random matrices Wishart matrix Tracy–Widom fluctuations asymptotic independence Bessel kernel

Citation

Hachem, Walid; Hardy, Adrien; Najim, Jamal. Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges. Ann. Probab. 44 (2016), no. 3, 2264--2348. doi:10.1214/15-AOP1022. http://projecteuclid.org/euclid.aop/1463410043.


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