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We establish the existence of a cutoff phenomenon for a natural analogue of the Brownian motion on free orthogonal quantum groups. We compute in particular the cutoff profile, whose type is different from the previously known examples and involves free Poisson laws and the semi-circle distribution. We prove convergence in total variation (and even in L p -norm for all p greater than 1) at times greater than the cutoff time and convergence in distribution for smaller times. We also study a similar process on quantum permutation groups, as well as the quantum random transposition walk. The latter yields in particular a quantum analogue of a recent result of the second-named author on random transpositions. 2020 Mathematics Subject Classification. 46L53, 60G10, 20G42. Key words and phrases. Cut-off phenomenon, Lévy process, quantum groups, random transpositions. 1 We do here (and sometimes in the sequel) a common abuse of notations, not writing the sequence indices.

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We consider random walks on vertex-transitive graphs of bounded degree. We show that subject to a simple diameter condition (which guarantees in particular that the walk is in some sense locally transient), the cover time fluctuations are universal: after rescaling, they converge to a standard Gumbel distribution. We further show by constructing an explicit counter-example that our diameter condition is sharp in a very strong sense. Surprisingly, this counter-example is also locally transient. We complement our result by showing that near the cover time, the distribution of the uncovered set is close in total variation to a product measure. The arguments rely on recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's theorem on groups of polynomial growth, which are leveraged into strong heat kernel bounds that imply decorrelation of the uncovered set. Another key aspect is an improvement on the exponential approximation of hitting times due to Aldous and Brown, which is of independent interest.