Hypercurrents

TL;DR

This paper introduces the concept of protocols on CW complexes, linking topological and analytical hypercurrents, and explores their properties and examples, especially in low temperature limits.

Contribution

It defines the hypercurrent associated with protocols on CW complexes and establishes the connection between algebraic topological and analytical hypercurrents.

Findings

Analytical hypercurrent converges to topological hypercurrent at low temperatures.

Examples of protocols with nontrivial hypercurrent are provided.

The notion of a protocol generalizes continuous-time Markov chains on manifolds.

Abstract

We introduce the notion of a protocol, which consists of a space whose points are labeled by real numbers indexed by the set of cells of a fixed CW complex in prescribed degrees, where the labels are required to vary continuously. If the space is a one-dimensional manifold, then a protocol determines a continuous time Markov chain. When a homological gap condition is present, we associate to each protocol a 'characteristic' cohomology class which we call the hypercurrent. The hypercurrent comes in two flavors: one algebraic topological and the other analytical. For generic protocols we show that the analytical hypercurrent tends to the topological hypercurrent in the low temperature limit. We also exhibit examples of protocols having nontrivial hypercurrent.