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We analyse the limiting behavior of the eigenvalue and singular valuedistribution for random convolution operators on large (not necessarilyAbelian) groups, extending the results by M. Meckes for the Abelian case. Weshow that for regular sequences of groups the limiting distribution ofeigenvalues (resp. singular values) is a mixture of eigenvalue (resp. singularvalue) distributions of Ginibre matrices with the directing measure beingrelated to the limiting behavior of the Plancherel measure of the sequence ofgroups. In particular for the sequence of symmetric groups, the limitingdistributions are just the circular and quarter circular laws, whereas e.g. forthe dihedral groups the limiting distributions have unbounded supports but aredifferent than in the Abelian case.  We also prove that under additional assumptions on the sequence of groups (inparticular for symmetric groups of increasing order) families of stochasticallyindependent random projection operators converge in moments to free circularelements.  Finally, in the Gaussian case we provide Central Limit Theorems for lineareigenvalue statistics.

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