Clifford algebras, combinatorics, and stochastic processes

Clifford algebras have been applied extensively in physics and engineering but have not been widely used in combinatorics. Moreover while fermionic stochastic processes have been studied, stochastic processes on Clifford algebras of arbitrary signature---which contain the real-, complex-, quaternion-valued and fermionic cases---have not. In the first half of this work, Clifford-algebraic methods are applied to combinatorics by creating Clifford adjacency matrices associated with finite graphs and Clifford stochastic matrices associated with Markov chains. These matrices reveal information about self-avoiding paths and self-avoiding stochastic processes on finite graphs and allow us to compute the expected number of Hamilton circuits in a random graph. In the second half of this work, stochastic processes on Clifford algebras are defined and specific examples, including Markov chains and Poisson processes, are constructed. We prove the existence of Clifford-algebraic stochastic integrals on the product space [0, t]m and utilize the graph-theoretic methods developed in Part I to recover the iterated stochastic integral by considering the limit in mean of a sequence of Berezin integrals of traces of matrices associated with finite graphs. As corollaries of known results, Hermite and Poisson-Charlier polynomials are recovered in this manner.