  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 $N\ln(N)/2(1-\cos(\theta))$. 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.
