  We show that the mixing times of random walks on compact groups can be used
to obtain concentration inequalities for the respective Haar measures. As an
application, we derive a concentration inequality for the empirical
distribution of eigenvalues of sums of random hermitian matrices, with possible
applications in free probability. The advantage over existing techniques is
that the new method can deal with functions that are non-Lipschitz or even
discontinuous with respect to the usual metrics.
