The ensemble of random Markov matrices

The ensemble of random Markov matrices is introduced as a set of Markov or stochastic matrices with the maximal Shannon entropy. The statistical properties of the stationary distribution pi, the average entropy growth rate $h$ and the second largest eigenvalue nu across the ensemble are studied. It is shown and heuristically proven that the entropy growth-rate and second largest eigenvalue of Markov matrices scale in average with dimension of matrices d as h ~ log(O(d)) and nu ~ d^(-1/2), respectively, yielding the asymptotic relation h tau_c ~ 1/2 between entropy h and correlation decay time tau_c = -1/log|nu| . Additionally, the correlation between h and and tau_c is analysed and is decreasing with increasing dimension d.