Graph-combinatorial approach for large deviations of Markov chains

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.

Automated summary

In this paper, we study the large deviations of discrete-time Markov chains through the pair empirical occupation measure. We provide exact expressions for the finite-time moment generating function and scaled cumulant generating function using a graph-combinatorial approach. These expressions enable a physical interpretation of different terms and may serve as a starting point for sub-leading asymptotic analysis. We illustrate the method using a simple two-state Markov chain.