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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.
