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

TL;DR

This paper introduces a one-parameter deformation of Wishart-Laguerre random matrix ensembles that results in fat-tailed spectral distributions, providing analytical insights and demonstrating universality and applicability to financial data.

Contribution

It presents a novel deformed Wishart-Laguerre model with power-law tails, deriving spectral properties analytically for finite and large matrices, and connecting to empirical financial data.

Findings

Derived generalized Marcenko-Pastur distribution with power-law tails.

Obtained universal generalized Bessel-law at the hard edge.

Showed good agreement with financial covariance matrix data.

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.