Sampling rare fluctuations of discrete-time Markov chains

TL;DR

This paper introduces a straightforward sampling method for rare fluctuations in discrete-time Markov chains, deriving large-deviation rate functions and demonstrating its application on models including active matter with phase separation.

Contribution

The paper presents a new simple method for sampling rare fluctuations in Markov chains with steady states, along with explicit expressions for large-deviation rate functions.

Findings

Effective sampling of rare fluctuations demonstrated

Derived bounds on large-deviation rate functions

Applied method to active matter model showing phase separation

Abstract

We describe a simple method that can be used to sample the rare fluctuations of discrete-time Markov chains. We focus on the case of Markov chains with well-defined steady-state measures, and derive expressions for the large-deviation rate functions (and upper bounds on such functions) for dynamical quantities extensive in the length of the Markov chain. We illustrate the method using a series of simple examples, and use it to study the fluctuations of a lattice-based model of active matter that can undergo motility-induced phase separation.