Random right eigenvalues of Gaussian quaternionic matrices

TL;DR

This paper proves that the empirical distribution of right eigenvalues of large Gaussian quaternionic matrices converges to a specific measure on the quaternionic unit ball, using potential theory.

Contribution

It establishes the almost sure convergence of eigenvalue distributions for Gaussian quaternionic matrices, extending random matrix theory to quaternions.

Findings

Eigenvalues concentrate on the quaternionic unit ball as matrix size increases

Empirical eigenvalue distribution converges to a specific measure

Provides insights into quaternionic random matrix models

Abstract

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance $1/n$. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension $n$ goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on more general Gaussian quaternionic random matrix models are also made.