In this paper we study (asymptotic) properties of the * -distribution of irreducible characters of finite quantum groups. We proceed in two steps, first examining the representation theory to determine irreducible representations and their powers, then we study the * -distribution of their trace with respect to the Haar measure. For the Sekine family we look at the asymptotic distribution (as the dimension of the algebra goes to infinity).

In this paper we study convergence of random walks, on finite quantum groups, arising from linear combination of irreducible characters. We bound the distance to the Haar state and determine the asymptotic behavior, i.e. the limit state if it exists. We note that the possible limits are any central idempotent state. We also look at cut-off phenomenon in the Sekine finite quantum groups. Keywords: convergence of random walks and finite quantum group and Sekine quantum groups and central idempotent state and representation theory

In this note we study asymptotic properties of the * -distribution of traces of some matrices, with respect to the free Haar trace on the unitary dual group. The considered matrices are powers of the unitary matrix generating the Brown algebra. We proceed in two steps, first computing the free cumulants of any R-cyclic family, then characterizing the asymptotic *distributions of the traces of powers of the generating matrix, thanks to these free cumulants. In particular, we obtain that these traces are asymptotic * -free circular variables. RésuméAspects stochastiques du groupe dual unitaire Dans cette note, nous étudions la loi asymptotique de la trace de certaines matrices, par rapport à la trace de Haar libre sur le groupe dual unitaire. Ces matrices sont les puissances de la matrice unitaire qui engendre l'algèbre de Brown. Nous procédons en deux étapes. Tout d'abord, nous calculons les cumulants joints d'une famille de matrices R-cyclique. Nous caractérisons ensuite la * -distributions asymptotique des traces considérées, à l'aide des cumulants libres. En particulier, nous obtenons que ces traces sont des variables asymptotiquement circulaires et * -libres.