Inter-relationships between orthogonal, unitary and symplectic matrix ensembles

TL;DR

This paper classifies specific weight functions that cause alternate eigenvalues from orthogonal, symplectic, and unitary matrix ensembles to form new ensembles, and analyzes their eigenvalue distributions.

Contribution

It provides a complete classification of weight functions linking orthogonal, symplectic, and unitary ensembles through eigenvalue decimation.

Findings

Classified weight functions for symplectic ensembles from orthogonal ensembles

Classified weight functions for unitary ensembles from unions of orthogonal ensembles

Analyzed k-point distributions for decimated orthogonal ensembles

Abstract

We consider the following problem: When do alternate eigenvalues taken from a matrix ensemble themselves form a matrix ensemble? More precisely, we classify all weight functions for which alternate eigenvalues from the corresponding orthogonal ensemble form a symplectic ensemble, and similarly classify those weights for which alternate eigenvalues from a union of two orthogonal ensembles forms a unitary ensemble. Also considered are the $k$-point distributions for the decimated orthogonal ensembles.