  To a unitary matrix U we associate a doubly stochastic matrix M by taking the
modulus squared of each element of U. To study the connection between onset of
quantum chaos on graphs and ergodicity of the underlying Markov chain,
specified by M, we study the limiting distribution of the spectral gap of M
when U is taken from the Circular Unitary Ensemble and the dimension N of U is
taken to infinity. We prove that the limiting distribution is degenerate: the
gap tends to its maximal value 1. The shape of the gap distribution for finite
N is also discussed.
