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

TL;DR

This paper discovers new equalities that reveal symmetries in cycle formation times of Markov chains and uses these to analyze circulation fluctuations, with implications for physics and biochemistry.

Contribution

It introduces a series of equalities characterizing cycle symmetry in Markov chains and applies these to study circulation fluctuations and large deviation principles.

Findings

Cycle formation times exhibit symmetry characterized by new equalities.

Empirical circulations follow a large deviation principle with a symmetric rate function.

Applications extend to statistical physics and biochemistry.

Abstract

In probability theory, equalities are much less than inequalities. 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 empirical circulations of a family of cycles passing through a common state satisfy a large deviation principle with a rate function which has an highly non-obvious symmetry. Finally, we discuss the applications of our work in statistical physics and biochemistry.