A note on the asymptotic uniformity of Markov chains with random rates

TL;DR

This work rigorously proves the rattling phenomenon for a broad class of random continuous-time Markov chains with i.i.d. transition rates, showing that the stationary distribution is effectively determined by simple local rate information. By combining concentration of exit rates, a perturbation framework for Markov chains, and a constructive approximation to a doubly stochastic matrix, the authors prove that the jump-chain stationary distribution π_P and the rate-based distribution ν_Q converge to the uniform distribution, and that π_Q is close to ν_Q. The main results quantify convergence in total variation with explicit rates: ∥ν_Q − u_n∥_1 = O(r_n) and ∥π_Q − u_n∥_1 = O(r_n/ε_n) under tail conditions (H), with r_n = sqrt((log n)/n) and ε_n controlling small-probability events. This provides a rigorous foundation for the rattling theory in non-equilibrium steady states and links to prior work by Chvykov and colleagues and Bordenave–Caputo–Chafaï, establishing a first broad class of systems where simple local rate functions predict long-run behavior.

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 Chafaï (2012) under certain assumptions.

Figures (2)

• Figure 1: Comparison of $\pi_Q$, $\nu_Q$, and the uniform distribution $u_n$ for Markov chains with i.i.d. transition rates. (a,b) Values of $n \pi_Q (i)$ are placed in descending order and plotted with the corresponding values of $n \nu_Q (i)$, for $n = 5000$. The dashed line represents the uniform distribution. The transition rates are either (a) standard exponential or (b) log-normal with unit variance. (c) The distribution $\pi_Q$ is closer in total variation distance to $\nu_Q$ than to uniform in each case, but both distances decrease as $n$ increases from $50$ to $5000$. The solid and dashed lines represent averages over $10$ independent trials, while the shaded regions represent these averages $\pm 1$ standard deviation.
• Figure 2: Non-uniformity of Markov chains with i.i.d. heavy-tailed transition rates. (a--c) The simulation parameters are the same as those in \ref{['fig: summary']}, except that the rates are equal in distribution to $X^{-1/\alpha}$, where $X$ is a standard exponential random variable, and either (a) $\alpha = 0.95$ or (b) $\alpha = 0.5$. Note that if $F$ is the cumulative distribution function of $X^{-1/\alpha}$, then $1 - F(x) \sim x^{-\alpha}$.

Theorems & Definitions (16)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1: Theorem 3.1 of Mitrophanov2005
• Proposition 2.2
• Proposition 2.3
• Remark 2.4
• Proposition 2.5
• proof : Proof of \ref{['thm:pip']}
• proof : Proof of \ref{['prop:concentration']}
• Proposition 3.1