Coherence and Uniqueness Theorems for Averaging Processes in Statistical Mechanics

Let S be the set of scalings {n^–1:n=1,2,3,...} and let L_z=zZ², zS, be the corresponding set of scaled lattices in R². In this paper averaging operators are defined for plaquette functions on L_z to plaquette functions on L_z for all z, zS, z=dz, d{2,3,4,...}, 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=1,2,3,... 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.