Varadhan's decomposition of shift-invariant closed uniform forms for large scale interacting systems on general crystal lattices

TL;DR

This paper establishes a generalized uniform Varadhan decomposition for shift-invariant closed forms in large-scale interacting systems on general crystal lattices, extending previous results to broader lattice structures.

Contribution

It generalizes Varadhan's decomposition to systems on general crystal lattices, including Euclidean lattices, beyond transferable graphs.

Findings

Proves a uniform version of Varadhan's decomposition for shift-invariant forms.

Extends previous results to general crystal lattices and Euclidean lattices.

Lays groundwork for future decomposition of closed L^2-forms in these systems.

Abstract

We prove a uniform version of Varadhan decomposition for shift-invariant closed uniform forms associated to large scale interacting systems on general crystal lattices. In particular, this result includes the case of translation invariant processes on Euclidean lattices $\mathbf{Z}^d$ with finite range. Our result generalizes the result of arXiv:2009.04699 which was valid for systems on transferable graphs. In subsequent research, we will use the result of this article to prove Varadhan's decomposition of closed $L^2$-forms for large scale interacting systems on general crystal lattices.