Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices

Publisher Website
Google Scholar
Abstract
To a unitary matrix U we associate a doubly stochastic matrix M by taking the squared modulus 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.