The ensemble of random Markov matrices
Abstract
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 tauc ~ 1/2 between entropy h and correlation decay time tauc = -1/log|nu| . Additionally, the correlation between h and and tauc is analysed and is decreasing with increasing dimension d.
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