Power-law deformation of Wishart-Laguerre ensembles of random matrices

Abstract

We introduce a one-parameter deformation of the Wishart-Laguerre or chiral ensembles of positive definite random matrices with Dyson index beta=1,2 and 4. Our generalised model has a fat-tailed distribution while preserving the invariance under orthogonal, unitary or symplectic transformations. The spectral properties are derived analytically for finite matrix size NxM for all three beta, in terms of the orthogonal polynomials of the standard Wishart-Laguerre ensembles. For large-N in a certain double scaling limit we obtain a generalised Marcenko-Pastur distribution on the macroscopic scale, and a generalised Bessel-law at the hard edge which is shown to be universal. Both macroscopic and microscopic correlations exhibit power-law tails, where the microscopic limit depends on beta and the difference M-N. In the limit where our parameter governing the power-law goes to infinity we recover the correlations of the Wishart-Laguerre ensembles. To illustrate these findings the generalised Marcenko-Pastur distribution is shown to be in very good agreement with empirical data from financial covariance matrices.