Power law deformation of Wishart–Laguerre ensembles of random matrices

We introduce a one-parameter deformation of the Wishart-Laguerre or chiralensembles of positive definite random matrices with Dyson index beta=1,2 and 4.Our generalised model has a fat-tailed distribution while preserving theinvariance under orthogonal, unitary or symplectic transformations. Thespectral properties are derived analytically for finite matrix size NxM for allthree beta, in terms of the orthogonal polynomials of the standardWishart-Laguerre ensembles. For large-N in a certain double scaling limit weobtain a generalised Marcenko-Pastur distribution on the macroscopic scale, anda generalised Bessel-law at the hard edge which is shown to be universal. Bothmacroscopic and microscopic correlations exhibit power-law tails, where themicroscopic limit depends on beta and the difference M-N. In the limit whereour parameter governing the power-law goes to infinity we recover thecorrelations of the Wishart-Laguerre ensembles. To illustrate these findingsthe generalised Marcenko-Pastur distribution is shown to be in very goodagreement with empirical data from financial covariance matrices.