We analyze the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by Meckes for the Abelian case. We show that for regular sequences of groups, the limiting distribution of eigenvalues (respectively singular values) is a mixture of eigenvalue (respectively singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular, for the sequence of symmetric groups, the limiting distributions are just the circular and quarter circular laws, whereas e.g. for the dihedral groups, the limiting distributions have unbounded supports but are different than in the Abelian case. We also prove that under additional assumptions on the sequence of groups (in particular, for symmetric groups of increasing order) families of stochastically independent random convolution operators converge in moments to free circular elements. Finally, in the Gaussian case, we provide Central Limit Theorems for linear eigenvalue statistics.

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随机非阿贝尔 G 循环矩阵。大型有限群上的随机卷积算子谱

我们分析了大型（不一定是阿贝尔）组上随机卷积算子的特征值和奇异值分布的限制行为，扩展了 Meckes 对阿贝尔情况的结果。我们表明，对于规则的组序列，特征值（分别为奇异值）的极限分布是 Ginibre 矩阵的特征值（分别为奇异值）分布的混合，其指导测量与序列的 Plancherel 测量的极限行为有关的组。特别是，对于对称群的序列，极限分布只是圆形和四分之一圆形定律，而例如对于二面体群，极限分布具有无界支持，但与阿贝尔情况不同。我们还证明，在对组序列的附加假设下（特别是对于递增顺序的对称组），随机独立的随机卷积算子族在瞬间收敛以释放圆形元素。最后，在高斯情况下，我们提供了线性特征值统计的中心极限定理。