The ne plus ultra of tree graph inequalities

Abstract

With nary mention of a tree graph, we obtain a cluster expansion bound that includes and vastly generalizes bounds as obtained by extant tree graph inequalities. This includes applications to both two-body and many-body potential situations of the recently obtained ‘new improved tree graph inequalities’ that have led to the ‘extra 1/ N ! factors’. We work in a formalism coupling a discrete set of boson variables, such as occurs in a lattice system in classical statistical mechanics, or in Euclidean quantum field theory. The estimates of this Letter apply to numerical factors as arising in cluster expansions, due to essentially arbitrary sequences of the basic operations: interpolation of the covariance, interpolation of the interaction, and integration by parts. This includes complicated evolutions, such as the repeated use of interpolation to decouple the same variables several times, to ensure higher connectivity for renormalization purposes, in quantum field theory.