Martin boundaries of the duals of free unitary quantum groups

Abstract

Given a free unitary quantum group G = Au(F), with F not a unitary 2-by-2 matrix, we show that the Martin boundary of the dual of G with respect to any G-Gˆ-invariant, irreducible, finite range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.