Coherence and uniqueness theorems for averaging processes in statistical mechanics

TL;DR

This paper establishes coherence and uniqueness theorems for averaging operators across scaled lattices in statistical mechanics, generalizing previous work and extending results to any dimension with a focus on the weights used.

Contribution

It introduces generalized averaging operators for all scales and dimensions, proving their coherence and the uniqueness of weights that produce such families.

Findings

Coherent families of averaging operators exist for any dimension.

Uniqueness theorems identify the only weights that produce coherence.

Generalization of previous averaging operators to higher dimensions.

Abstract

Let S be the set of scalings 1, 2,3,4, ... and consider the corresponding set of scaled lattices in the plane. In this paper averaging operators are defined for plaquette functions on a lattice to plaquette functions on a coarser lattice for all scale factors and their coherence is proved. This generalizes the averaging operators introduced by Balaban and Federbush. There are such coherent families of averaging operators for any dimension D and not only for D=2. Finally there are uniqueness theorems saying that in a sense, besides a form of straightforward averaging, the weights used are the only ones that give coherent families of averaging operators.