Cutoff profiles for quantum Lévy processes and quantum random transpositions

Abstract

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.

Random Walks on the Symmetric Group: Cutoff for One-sided Transposition Shuffles

“…In order to prove the limit profile of the random transposition shuffle Teyssier derived a improvement of Diaconis' upper bound lemma (Lemma 1.2.35). This improvement of the upper bound lemma has opened the path to study limit profiles of many other random walks, including the previously mentioned random k-cycles shuffle [21,35]. This collection of work demonstrates how important the random transposition shuffle is to the study of random walks on groups and how it is still influencing the field even today.…”

A state-space approach to quantum permutations

“…The quantum transposition ϕ tr studied by Freslon, Teyssier and Wang [20] is a central state, and it is the only central quantum transposition in S + N . Central states such as ϕ tr have some nice properties: that for any irreducible representation n ∈ N ≥0 , ϕ tr (ρ (n) ij ) = ϕ tr (n)δ i,j , and as the matrix elements of the irreducible representations form a basis of C(S + N ), they are completely determined by their restriction to the central algebra C(S + N ) 0 generated by the characters as:…”