Order of magnitude time-reversible Markov chains and characterization of clustering processes

TL;DR

This paper introduces the concept of order of magnitude reversibility in Markov chains, characterizes clustering processes in biological systems, and analyzes their stationary distributions in the limit of small parameters.

Contribution

It defines OM-reversibility, proves a detailed balance analogue, and applies these results to clustering phenomena in interacting particle systems.

Findings

OM-reversibility is a natural property in macroscopic systems.

Clustering processes are OM-reversible with explicit stationary distribution orders.

States with all particles clustered dominate as diffusion rate approaches zero.

Abstract

We introduce the notion of order of magnitude reversibility (OM-reversibility) in Markov chains that are parametrized by a positive parameter $\ep$. OM-reversibility is a weaker condition than reversibility, and requires only the knowledge of order of magnitude of the transition probabilities. For an irreducible, OM-reversible Markov chain on a finite state space, we prove that the stationary distribution satisfies order of magnitude detailed balance (analog of detailed balance in reversible Markov chains). The result characterizes the states with positive probability in the limit of the stationary distribution as $\ep \to 0$, which finds an important application in the case of singularly perturbed Markov chains that are reducible for $\ep=0$. We show that OM-reversibility occurs naturally in macroscopic systems, involving many interacting particles. Clustering is a common phenomenon in biological systems, in which particles or molecules aggregate at one location. We give a simple condition on the transition probabilities in an interacting particle Markov chain that characterizes clustering. We show that such clustering processes are OM-reversible, and we find explicitly the order of magnitude of the stationary distribution. Further, we show that the single pole states, in which all particles are at a single vertex, are the only states with positive probability in the limit of the stationary distribution as the rate of diffusion goes to zero.