Open Access
2016 Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge
Walid Hachem, Adrien Hardy, Jamal Najim
Electron. J. Probab. 21: 1-36 (2016). DOI: 10.1214/15-EJP4441

Abstract

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.

Citation

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Walid Hachem. Adrien Hardy. Jamal Najim. "Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge." Electron. J. Probab. 21 1 - 36, 2016. https://doi.org/10.1214/15-EJP4441

Information

Received: 2 January 2013; Accepted: 17 December 2014; Published: 2016
First available in Project Euclid: 3 February 2016

zbMATH: 1336.15016
MathSciNet: MR3485343
Digital Object Identifier: 10.1214/15-EJP4441

Subjects:
Primary: 15A52
Secondary: 15A18‎ , 60F15

Keywords: Bessel kernel , large random matrices , Pearcey kernel , Wishart matrix

Vol.21 • 2016
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