Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

Abstract

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.