Tridiagonal realization of the anti-symmetric Gaussian $\beta$-ensemble

TL;DR

This paper derives explicit distributions for eigenvalues and eigenvectors of anti-symmetric Gaussian beta-ensemble matrices using three different mathematical proofs, enhancing understanding of their spectral properties.

Contribution

It provides three novel proofs for the distribution of eigenvalues and eigenvectors of anti-symmetric Gaussian beta-ensemble matrices, including explicit Jacobian calculations and mappings to Laguerre ensembles.

Findings

Eigenvalue density functions are explicitly computed.

Distribution of first eigenvector components is derived.

Connections to Laguerre beta-ensemble are established.

Abstract

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pr\"ufer phases of the random matrices.