## Electronic Journal of Probability

### Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 1, 36 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1454514661

Digital Object Identifier
doi:10.1214/15-EJP4441

Mathematical Reviews number (MathSciNet)
MR3485343

Zentralblatt MATH identifier
1336.15016

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

#### Citation

Hachem, Walid; Hardy, Adrien; Najim, Jamal. Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge. Electron. J. Probab. 21 (2016), paper no. 1, 36 pp. doi:10.1214/15-EJP4441. https://projecteuclid.org/euclid.ejp/1454514661

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