Power law deformation of Wishart–Laguerre ensembles of random matrices

Abstract

We introduce a one-parameter deformation of the Wishart–Laguerre orchiral ensembles of positive definite random matrices with Dyson indexβ = 1,2 and 4. Our generalized model has a fat-tailed distribution while preservingthe invariance under orthogonal, unitary or symplectic transformations.The spectral properties are derived analytically for finite matrix sizeN × M for all threevalues of β, in terms of the orthogonal polynomials of the standard Wishart–Laguerre ensembles. For largeN in a certain double-scaling limit we obtain a generalized Marčenko–Pasturdistribution on the macroscopic scale, and a generalized Bessel law at thehard edge which is shown to be universal. Both macroscopic and microscopiccorrelations exhibit power law tails, where the microscopic limit depends onβ and thedifference M−N. In the limit where our parameter governing the power law goes to infinity we recover thecorrelations of the Wishart–Laguerre ensembles. To illustrate these findings, the generalizedMarčenko–Pastur distribution is shown to be in very good agreement with empirical datafrom financial covariance matrices.