Random symmetric matrices on Clifford algebras

TL;DR

This paper studies stochastic processes on symmetric matrices derived from Clifford algebras, revealing spectral measure behaviors and connecting to Bott's periodicity.

Contribution

It introduces a framework for analyzing spectral measures of matrix processes from Clifford algebras, linking algebraic structures to spectral properties.

Findings

Identification of eigenvalue multiplicities

Spectral measure processes characterized

Connection to Bott's periodicity established

Abstract

We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we thus recover Bott's periodicity.