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May 2016 Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges
Walid Hachem, Adrien Hardy, Jamal Najim
Ann. Probab. 44(3): 2264-2348 (May 2016). DOI: 10.1214/15-AOP1022


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 $N^{2}$.


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Walid Hachem. Adrien Hardy. Jamal Najim. "Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges." Ann. Probab. 44 (3) 2264 - 2348, May 2016.


Received: 1 September 2014; Revised: 1 February 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1346.15035
MathSciNet: MR3502605
Digital Object Identifier: 10.1214/15-AOP1022

Primary: 15A52
Secondary: 15A18‎ , 60F15

Keywords: Asymptotic independence , Bessel kernel , large random matrices , Tracy–Widom fluctuations , Wishart matrix

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • May 2016
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