Dyson processes on the octonion algebra

TL;DR

This paper explores Brownian motion on octonion matrices, analyzing spectral laws and the impact of octonion nonassociativity on eigenvalue behavior, revealing unique spectral properties linked to matrix dimension.

Contribution

It introduces two models of Brownian motion on octonion matrices and investigates how nonassociativity influences eigenvalue multiplicities and spectral laws.

Findings

Eigenvalue multiplicity relates to the exponent in the spectral law.

Nonassociativity of octonions affects spectral properties.

Matrix dimension plays a crucial role in spectral behavior.

Abstract

We consider Brownian motion on symmetric matrices of octonions, and study the law of the spectrum. Due to the fact that the octonion algebra is nonassociative, the dimension of the matrices plays a special role. We provide two specific models on octonions, which give some indication of the relation between the multiplicity of eigenvalues and the exponent in the law of the spectrum.