Stochastic aspects of the unitary dual group

Abstract

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.

On some high-dimensional limits of matricial stochastic processes seen from a quantum probability perspective

“…Dual groups have been studied somewhat, though less than quantum groups. For instance, one can refer to [23,9,21,5,13,20,2,3]. In particular, in [20], the author extended a previous result due to Biane in [4].…”

“…• We remark that the (1)-part is the one due to the quadratic variation (a), the (2) part is due to (c), and the (3) part is due to (c). So, similarly to the orthogonal case, we end up with seven operators: D, U (1) , U (2) , U (3) , B (1) , B (2) and B (3) . Again, we will be interested only in the m → ∞ limit of these operators.…”

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

“…Furthermore, inspired by computations in (Kirillov, 2004) page 59-61 about the Plancherel measure in the unitary of the 3-dimensional Heisenberg Lie group, we shall generalize this result to the Plancherel measure in the unitary dual of -dimensional Heisenberg Lie group corresponding to its irreducible unitary representation presented in (Corwin & Greenleaf, 1990) page 48. In the previous work we can see the notions and some applications of unitary dual groups (Bagarello & Russo, 2018), (Baraquin, 2017), and (Cebron & Ulrich, 2016).…”