On the semicircular law of large dimensional random quaternion matrices

TL;DR

This paper proves that the empirical spectral distribution of large quaternion self-dual Hermitian matrices converges to the semicircular law, extending universality beyond Gaussian ensembles.

Contribution

It establishes the semicircular law for large quaternion matrices without assuming Gaussian distribution, advancing understanding of spectral universality.

Findings

Empirical spectral distribution converges to semicircular law

Universal behavior observed beyond Gaussian assumptions

Provides a key lemma for spectral analysis

Abstract

It is well known that Gaussian symplectic ensemble (GSE) is defined on the space of $n\times n$ quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices. As a natural idea we want to get more universal results by removing the Gaussian condition. For the first step, in this paper we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.