Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices

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Abstract
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.

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