Ensemble inter-relations in random matrix theory

Abstract

The ensemble inter-relations to be considered are special features of classical cases, where the joint eigenvalue probability density can be computed explicitly. Attention will be focussed too on the consequences of these inter-relations, most often in relation to gap probabilities. A highlight from this viewpoint is an evaluation formula for the gap probability generating function for the circular orthogonal and circular symplectic ensembles (examples of Pfaffian point processes), in terms of the gap probability generating function for real orthogonal matrices chosen with Haar measure ensembles (examples of determinantal point processes). The classes of inter-relations which lead to this result involve the superposition of the eigenvalue point processes of two independent classical ensembles, or consideration of the singular values of the classical ensembles with an evenness symmetry, and the decimation operation of integrating over every second eigenvalue. It turns out that the probability density functions encountered through the consideration of superposition and decimation can in some cases be obtained via the consideration of various rank one matrix perturbations. These allow for different understandings of some of the inter-relations in this class. We make note too of inter-relations -- which we consider generally as an identity or evaluation formula of a statistic linking two distinct random matrix ensemble -- that fall outside of this class, applying to the spectral form factor, and ones relating to discrete determinantal point processes. Not part of our review are the so-called duality relations of random matrix theory. Their consideration warrants a separate discussion.