# Stochastic aspects of the unitary dual group

## TL;DR

This paper investigates the asymptotic *-distribution of traces of powers of a unitary matrix within the free probability framework, revealing they behave as free circular variables in the limit.

## Contribution

It provides a detailed analysis of the asymptotic *-distributions of traces of powers of a generating matrix in the free Haar setting, using free cumulants.

## Key findings

- Traces of powers are asymptotic *-free circular variables.
- Computed free cumulants for R-cyclic families.
- Characterized asymptotic *-distributions of matrix traces.

## 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.