随机马尔可夫矩阵的系综
The ensemble of random Markov matrices
摘要：
随机马尔可夫矩阵系综被引入作为一组具有最大Shannon熵的马尔可夫或随机阵。研究了系综上的平稳分布pi、平均熵增长率$h$和第二大特征值nu的统计性质。本文证明了马氏矩阵的熵增长率和次大特征值分别与矩阵d的维数h ~ log（O（d））和nu ~ d^（-1/2）成平均比例，并给出了熵h与相关衰减时间τ_c之间的渐近关系h τ_c ~ 1/2 = -1/log |nu| .此外，还分析了h和τ_c之间的相关性，并发现h和τ_c之间的相关性随维数d的增加而减小。
英文摘要：
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.
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