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

Abstract: 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.