Muttalib–Borodin ensembles in random matrix theory — Realisations and correlation functions

Abstract

Muttalib–Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form Q (Formula presented). We study the Laguerre and Jacobi versions of this model — so named by the form of the one-body interaction terms — and show that for θ ∈ ℤ+ they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general θ > 0, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent result due to Cheliotis, although our derivation is different. We make use of a generalisation of a double contour integral formula for the correlatio..