Cut-off phenomenon for random walks on free orthogonal quantum groups

We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups. As a consequence, we prove that the analogue of the uniform plane Kac walk on this quantum group has a cut-off at $N\ln (N)\mathrm{/}2(1-\cos (\theta ))$. This is the first result of this type for genuine compact quantum groups. We also obtain similar results for mixtures of rotations and quantum permutations.