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We introduce the notion of order of magnitude reversibility(OM-reversibility) in Markov chains that are parametrized by a positiveparameter $\ep$. OM-reversibility is a weaker condition than reversibility, andrequires only the knowledge of order of magnitude of the transitionprobabilities. For an irreducible, OM-reversible Markov chain on a finite statespace, we prove that the stationary distribution satisfies order of magnitudedetailed balance (analog of detailed balance in reversible Markov chains). Theresult characterizes the states with positive probability in the limit of thestationary distribution as $\ep \to 0$, which finds an important application inthe 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 inbiological systems, in which particles or molecules aggregate at one location.We give a simple condition on the transition probabilities in an interactingparticle Markov chain that characterizes clustering. We show that suchclustering processes are OM-reversible, and we find explicitly the order ofmagnitude of the stationary distribution. Further, we show that the single polestates, in which all particles are at a single vertex, are the only states withpositive probability in the limit of the stationary distribution as the rate ofdiffusion goes to zero.

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