A random matrix decimation procedure relating $\beta = 2/(r+1)$ to $\beta = 2(r+1)$

TL;DR

This paper establishes a novel relationship between eigenvalue distributions of certain beta-ensembles, linking ensembles with parameters  = 2/(r+1) and 2(r+1) through a decimation procedure, extending classical symmetry results.

Contribution

It introduces a decimation procedure that relates  = 2/(r+1) and 2(r+1) beta-ensembles, generalizing classical symmetry relations in random matrix theory.

Findings

Joint distribution of every (r+1)-st eigenvalue in  = 2/(r+1) ensemble equals that of  = 2(r+1) ensemble.

Generalization of Dixon and Anderson's conditional probability density function.

Demonstrates inter-relations between eigenvalue distributions for generalized -ensembles.

Abstract

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $\beta$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $\beta$-ensembles with $\beta = 2/(r+1)$ is equal to that of another $\beta$-ensemble with $\beta = 2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.