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

• Hachem, Walid ^[2] ; Hardy, Adrien ^[1] ; Najim, Jamal ^[3]
  1. [1] Royal Institute of Technology

     Royal Institute of Technology

     Suecia

     
  2. [2] Télécom ParisTech
  3. [3] Université Paris-Est

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 3, 2016, págs. 2264-2348
• Idioma: inglés
• DOI: 10.1214/15-AOP1022
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• Resumen

  • 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 N2/3. 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 N2.