In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In [6], the first and second authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalizations: (i) we state results of freeness in a context of general sequences of representations of the unitary group — the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.

在本文中，我们继续研究具有张量结构的随机矩阵族的渐近性质。在 [6] 中，第一作者和第二作者提供了酉随机矩阵的张量积相对于归一化迹渐近自由的条件。在这里，我们通过证明 Haar 酉矩阵的张量积的渐近自由度适用于明显更大的状态类来扩展这一结果。我们的结果依赖于对称群下的不变性，因此依赖于流量概率。作为副产品，我们探索了两个额外的概括：