The ensemble of random Markov matrices

The ensemble of random Markov matrices is introduced as the set of Markov or stochastic matrices with the maximal Shannon entropy. Studied are statistical properties of the invariant distribution pi, the average entropy growth rate h and the second largest eigenvalue nu. It is shown and heuristically proved that in average the entropy growth rate and second largest eigenvalue of Markov matrices scales with dimension d as h ~ log(O(d)) and nu ~ d^(-1/2), respectively. Interestingly in Markov matrices h/log(|nu|) ~ 1/2 for large dimensions. Additionally the correlation between h and the reciprocal correlation decay time -log|nu|, averaged over the ensemble, is analysed and it is found to present but decreases with increasing dimension d.