Martin boundaries of the duals of free unitary quantum groups

TL;DR

This paper establishes that the Martin boundary of the dual of certain free unitary quantum groups matches the topological boundary previously defined, extending classical boundary concepts to a quantum setting.

Contribution

It proves the coincidence of the Martin boundary with the topological boundary for duals of free unitary quantum groups, a novel quantum analogue of classical results.

Findings

Martin boundary coincides with the topological boundary for these quantum groups

Quantum random walks exhibit boundary behavior similar to classical free groups

Extends classical boundary theory to quantum group duals

Abstract

Given a free unitary quantum group $G=A_u(F)$, with $F$ not a unitary $2$-by-$2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-$\hat G$-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.