Martin boundaries of the duals of free unitary quantum groups

Given a free unitary quantum group $G=A_{u}(F)$, with $F$ not a unitary $2\times 2$ 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.

Probabilistic boundaries of finite extensions of quantum groups

Given a discrete quantum group [Formula: see text] with a finite normal quantum subgroup [Formula: see text], we show that any positive, possibly unbounded, harmonic function on [Formula: see text] with respect to an irreducible invariant random walk is [Formula: see text]-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of [Formula: see text] coincide with those of [Formula: see text]. A similar result is also proved in the setting of exact sequences of C[Formula: see text]-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical.