Topological Structures of Large Scale Interacting Systems via Uniform Functions and Forms

TL;DR

This paper introduces uniform cohomology to analyze the topological structure of large scale interacting systems on infinite graphs, providing a new framework to identify macroscopic observables from microscopic details.

Contribution

It constructs a novel uniform cohomology theory using uniform functions, offering a new perspective on macroscopic observables and decomposing shift-invariant forms in large scale systems.

Findings

Proves a decomposition theorem for shift-invariant closed uniform forms.

Extends Varadhan's decomposition result to a more general setting.

Provides a foundation for analyzing hydrodynamic limits in interacting systems.

Abstract

In this article, we investigate the topological structure of large scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed $L^2$-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan's decomposition theorem for a general class of large scale interacting systems.