Quantum reflections, random walks and cut-off

Abstract

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of reflections. We also study random walks on quantum reflection groups and more generally on free wreath products of finite groups by quantum permutation groups.2010 Mathematics Subject Classification. 60J05, 60B15, 20G42.

Random Walks on Finite Quantum Groups

“…If n is odd, there are only four possibilities: (8) does not hold, i.e. µ = h Γ,l,τ with q = 1 (and p = n), l = 0 and τ i = 1 for all i ∈ Z n , • otherwise the random walk diverges.…”

“…• condition (8) holds, i.e. µ = KP n , (8) does not hold, i.e. µ = h Γ,l,τ with q = 1 (and p = n), l = 0 and τ i = 1 for all i ∈ Z n ,…”

“…In[12], McCarthy gives another example of a random walk in Sekine family where there is no cut-off, but with a state which formally does not depend on n and which is not the Fourier transform of an element in the central algebra. Note that in[7] and[8], Freslon finds cut-off examples in compact quantum groups.…”

The Ergodic Theorem for Random Walks on Finite Quantum Groups

“…As will be seen in Section 5.4, as the representation theory generalises so well from classical to quantum, the upper bound lemma of Diaconis and Shahshahani can also be used to analyse random walks on quantum groups. The upper bound lemma has been used to analyse random walks on the dual symmetric group, S n [27]; Sekine quantum groups, Y n [2,27]; the Kac-Paljutkin quantum group, G 0 [2]; free orthogonal quantum groups, O + N [17]; free symmetric quantum groups, S + N [17]; the quantum automorphism group of (M N (C), tr) [17]; free unitary groups, U + N [18]; free wreath products Γ ≀ * S + N , including quantum reflection groups H s + N [18]; duals of discrete groups, Γ, including for Γ = F N the free group on N generators, [19].…”

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups

“…However, the classical Diaconis-Shahshahani Theory has been extended to possiblyinfinite compact groups by Rosenthal [14]. Freslon [7,8] has also extended this work on finite quantum groups, developed during the author's PhD studies, to the far more technically-involved case of compact quantum groups, where the algebras are no longer necessarily finite dimensional. Freslon presents many interesting examples of random walks on compact quantum groups -with the cut-off phenomenon -and also addresses, particularly in [7], a number of concerns not answered or addressed in the author's PhD thesis.…”