Graph-combinatorial approach for large deviations of Markov chains

TL;DR

This paper introduces a graph-combinatorial method to analyze large deviations in Markov chains, providing exact formulas for moment generating functions and insights into fluctuation behaviors.

Contribution

It presents a novel graph-based approach to compute large deviation functions for pair empirical measures in Markov chains, including physical interpretations.

Findings

Exact expression for finite-time moment generating function

Decomposition into cycles and paths contributions

Application to a two-state Markov chain

Abstract

We consider discrete-time Markov chains and study large deviations of the pair empirical occupation measure, which is useful to compute fluctuations of pure-additive and jump-type observables. We provide an exact expression for the finite-time moment generating function, which is split in cycles and paths contributions, and scaled cumulant generating function of the pair empirical occupation measure via a graph-combinatorial approach. The expression obtained allows us to give a physical interpretation of interaction and entropic terms, and of the Lagrange multipliers, and may serve as a starting point for sub-leading asymptotics. We illustrate the use of the method for a simple two-state Markov chain.