On fluctuations of cycles in a finite CW complex

TL;DR

This paper applies algebraic topology to analyze stochastic cellular cycles in finite CW complexes, revealing that the average current quantizes fractionally in low-temperature limits, with denominators linked to combinatorial invariants.

Contribution

It introduces a homological observable called the average current and proves its fractional quantization in the low-temperature, adiabatic limit, connecting topology with stochastic dynamics.

Findings

Average current fractionally quantizes in the low-temperature limit

Quantization denominators are combinatorial invariants of the CW complex

Establishes a link between algebraic topology and stochastic process behavior

Abstract

We use algebraic topology to study the stochastic mo- tion of cellular cycles in a finite CW complex. Inspired by statis- tical mechanics, we introduce a homological observable called the average current. The latter measures the average flux of the prob- ability 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.