The ensemble of random Markov matrices

TL;DR

This paper introduces the ensemble of random Markov matrices with maximal Shannon entropy and studies their statistical properties, revealing how key metrics scale with matrix dimension and their interrelations.

Contribution

It provides a heuristic proof and analysis of the scaling laws of entropy growth rate and second largest eigenvalue in random Markov matrices, linking them to matrix dimension.

Findings

Entropy growth rate scales as log(d) with dimension d.

Second largest eigenvalue scales as d^(-1/2).

Entropy and correlation decay time are asymptotically related by h tau_c ~ 1/2.

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 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.