Topological structures of large-scale interacting systems via uniform functions and forms
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. Uni...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
|
| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000616/type/journal_article |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 2050-5094 |