Cluster expansion for abstract polymer models. New bounds from an old approach

TL;DR

This paper improves the convergence conditions for cluster expansions in abstract polymer models by refining classical tree graph methods, leveraging Penrose identities and iterated transformations to achieve better bounds.

Contribution

It introduces a new convergence criterion that surpasses previous bounds by Kotecky-Preiss and Dobrushin, based on a refined analysis of tree graphs and Penrose identities.

Findings

New convergence bounds for cluster expansions.

Examples demonstrating improved bounds.

Enhanced analytical techniques for polymer models.

Abstract

We revisit the classical approach to cluster expansions, based on tree graphs, and establish a new convergence condition that improves those by Kotecky-Preiss and Dobrushin, as we show in some examples. The two ingredients of our approach are: (i) a careful consideration of the Penrose identity for truncated functions, and (ii) the use of iterated transformations to bound tree-graph expansions.