Spherical Matrix Ensembles

TL;DR

This paper introduces spherical matrix ensembles with fixed Frobenius norm, derives their eigenvalue distributions, and shows their spectral measures converge to the semicircular law as matrix size increases, providing explicit formulas in some cases.

Contribution

It defines spherical orthogonal, unitary, and symplectic ensembles and analyzes their eigenvalue distributions, including explicit spectral density formulas for the unitary case.

Findings

Eigenvalue distributions are explicitly determined for each ensemble and matrix size.

Empirical spectral measures converge rapidly to the semicircular distribution as N increases.

Explicit spectral density formulas are provided for the unitary ensemble.

Abstract

The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_\beta(N,r)$ consist of $N \times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uniform measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N \to \infty$. In the unitary case ($\beta=2$), we also find an explicit formula for the empirical spectral density for each $N$.