  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 $N\to\infty$ 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.
