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 pure-additive jump-type observables. provide an exact expression for finite-time moment generating function, split in cycles paths contributions, scaled cumulant function measure via a graph-combinatorial approach. The obtained allows us give physical interpretation interaction entropic terms, Lagrange multipliers, may serve as starting point sub-leading asymptotics. illustrate use method simple two-state chain.