On the convergence of cluster expansions for polymer gases

TL;DR

This paper compares various convergence criteria for cluster expansions in polymer gases, demonstrating that a sharper criterion can be proven using simpler methods and extending the results to more general polymer models.

Contribution

It introduces a unified approach to establish convergence criteria for polymer gases, improving bounds and simplifying proofs for subset and abstract polymers.

Findings

The sharper convergence criterion can be proven via adapted Dobrushin and elementary Kirkwood-Salzburg methods.

The alternative treatment yields the same convergence region as Dobrushin's criterion.

The methods allow systematic improvements in bounds on correlations.

Abstract

We compare the different convergence criteria available for cluster expansions of polymer gases subjected to hard-core exclusions, with emphasis on polymers defined as finite subsets of a countable set (e.g. contour expansions and more generally high- and low-temperature expansions). In order of increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via a direct combinatorial handling of the terms of the expansion. We show that for subset polymers our sharper criterion can be proven both by a suitable adaptation of Dobrushin inductive argument and by an alternative --in fact, more elementary-- handling of the Kirkwood-Salzburg equations. In addition we show that for general abstract polymers this alternative treatment leads to the same convergence region as the inductive Dobrushin argument and, furthermore, to a systematic way to improve bounds on correlations.