  We analyse the limiting behavior of the eigenvalue and singular value
distribution for random convolution operators on large (not necessarily
Abelian) groups, extending the results by M. Meckes for the Abelian case. We
show that for regular sequences of groups the limiting distribution of
eigenvalues (resp. singular values) is a mixture of eigenvalue (resp. 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 projection operators converge in moments to free circular
elements.
  Finally, in the Gaussian case we provide Central Limit Theorems for linear
eigenvalue statistics.
