Fluctuations of cycles in a finite CW complex

Abstract

We use algebraic topology to study the stochastic motion of cellular cycles in a finite CW complex. Inspired by statistical mechanics, we introduce a homological observable called the average current. The latter measures the average flux of the probability in the process. In the low temperature, adiabatic limit, we prove that the average current fractionally quantizes, in which the denominators are combinatorial invariants of the CW complex.