Unitary stochastic matrix ensembles and spectral statistics

We propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with Markov processes on graphs. We argue that the spectral statistics of such an ensemble (after ensemble averaging) depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix. It is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size $N\to \mathrm{\infty }$ if the spectral gap of the corresponding transition matrices closes slower than 1/N. The hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non-uniformly connected star graphs and diffusive networks in arbitrary dimensions.