Martin boundary theory of some quantum random walks

Abstract

In this paper we define a general setting for Martin boundary theory associated to quantum random walks, and prove a representation theorem. We show that in the dual of a simply connected Lie subgroup of U( n), the extremal Martin boundary is homeomorphic to a sphere. Then, we investigate restriction of quantum random walks to Abelian subalgebras of group algebras, and establish a Ney–Spitzer theorem for an elementary random walk on the fusion algebra of SU( n), generalizing a previous result of Biane. We also consider the restriction of a quantum random walk on SU q ( n) introduced by Izumi to two natural Abelian subalgebras, and relate the underlying Markov chains by classical probabilistic processes. This result generalizes a result of Biane.