Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices

To a unitary matrix U we associate a doubly stochastic matrix M by taking t he squared modulus of each element of U. To study the connection between on set of quantum chaos on graphs and ergodicity of the underlying Markov chai n, specified by M, we study the limiting distribution of the spectral gap o f 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 degen erate: the gap tends to its maximal value 1. The shape of the gap distribut ion for finite N is also discussed.