Octonions in random matrix theory

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 matrices 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+X, when X has random octonion entries. Analytic results are obtained from N = 2, but are observed to break down in the 3 x 3 case.