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.
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Acknowledgements
The author is grateful to Cyril Houdayer and Maxime Février for discussions connected to the present work, as well as to the referees for their remarks and comments.
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Freslon, A. Cut-off phenomenon for random walks on free orthogonal quantum groups. Probab. Theory Relat. Fields 174, 731–760 (2019). https://doi.org/10.1007/s00440-018-0863-8
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DOI: https://doi.org/10.1007/s00440-018-0863-8