Cycle symmetries and circulation fluctuations for discrete-time and continuous-time Markov chains

Abstract

In this paper, we find a series of equalities which characterize the symmetry of the forming times of a family of similar cycles for discrete-time and continuous-time Markov chains. Moreover, we use these cycle symmetries to study the circulation fluctuations for Markov chains. We prove that the sample circulations along a family of cycles passing through a common state satisfy a large deviation principle with a rate function which has a highly nonobvious symmetry. Further extensions and applications to statistical physics and biochemistry are also discussed, especially the fluctuation theorems for the sample net circulations.