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.

中文翻译：

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酉对偶群的随机方面

在这篇笔记中，我们研究了一些矩阵的迹的 *-分布的渐近性质，相对于酉对偶群上的自由 Haar 迹。所考虑的矩阵是生成布朗代数的酉矩阵的幂。我们分两步进行，首先计算任何 R 循环族的自由累积量，然后表征生成矩阵的幂迹的渐近 * -分布，这要归功于这些自由累积量。特别是，我们得到这些轨迹是渐近 *-free 循环变量。