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In probability theory, equalities are much less than inequalities. In thispaper, we find a series of equalities which characterize the symmetry of theforming times of a family of similar cycles for discrete-time andcontinuous-time Markov chains. Moreover, we use these cycle symmetries to studythe circulation fluctuations for Markov chains. We prove that the empiricalcirculations of a family of cycles passing through a common state satisfy alarge deviation principle with a rate function which has an highly non-obvioussymmetry. Finally, we discuss the applications of our work in statisticalphysics and biochemistry.

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