Geodesic
Topological structures of large-scale interacting systems via uniform functions and forms
Forum of Mathematics, Sigma, Tome 12 (2024)

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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. 
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Kenichi Bannai; Yukio Kametani; Makiko Sasada. Topological structures of large-scale interacting systems via uniform functions and forms. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.61

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