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.

40 pages, 6 figures. Accepted for publication in EJP

References in corpus (5)

• The Pearcey Process
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices
• Finite N corrections to the limiting distribution of the smallest eigenvalue of Wishart complex matrices
• A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge
• A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices

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