A note on the asymptotic uniformity of Markov chains with random rates (2505.01608v1)

Abstract: The stationary distribution of a continuous-time Markov chain is generally a complicated function of its transition rates. However, we show that if the transition rates are i.i.d. random variables with a common distribution satisfying certain tail conditions, then the resulting stationary distribution is close in total variation distance to the distribution that is proportional to the inverse of the exit rates of the states. This result, which generalizes and makes a precise prediction of Chvykov et al. (2021), constitutes the first rigorous validation of an emerging physical theory of order in non-equilibrium systems. The proof entails showing that the stationary distribution of the corresponding "jump chain," i.e., the discrete-time Markov chain with transition probabilities given by the normalized transition rates, is asymptotically uniform as the number of states grows, which settles a question raised by Bordenave, Caputo, and Chafa\"i (2012) under certain assumptions.