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 π, the average entropy growth rate h and the second-largest eigenvalue ν across the ensemble are studied. It is shown and heuristically proven that the entropy growth rate and second-largest eigenvalue of Markov matrices scale on average with the dimension of the matrices d as h∼log(O(d)) and |ν|∼d−1/2, respectively, yielding the asymptotic relation hτc∼1/2 between the entropy h and the correlation decay time τ = −1/log|ν|. Additionally, the correlation between h and τc is analysed; it decreases with increasing dimension d