Octonions in random matrix theory

TL;DR

This paper develops an analytic theory for Hermitian random matrices with octonion entries, extending previous work from the N=2 case to N=3 using Jordan algebra, and explores the limitations of these results.

Contribution

It introduces a Jordan algebra approach to analyze N=3 Hermitian random matrices with octonion entries, advancing beyond the known N=2 case.

Findings

Analytic results for N=2 are obtained.

Results break down for N=3, indicating complexity.

Provides insights into octonion-based random matrix structures.

Abstract

The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random by symmetry considerations. Only for $N=2$ is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for $N=3$. We then proceed to consider the matrix structure $X^\dagger X$, when $X$ has random octonion entries. Analytic results are obtained from $N=2$, but are observed to break down in the $3 \times 3$ case.