Sufficient conditions for uniform bounds in abstract polymer systems and explorative partition schemes

TL;DR

This paper introduces new sufficient conditions for uniform bounds in abstract polymer systems, resolving discrepancies between previous conditions by extending the tree-operator approach and employing novel partition schemes.

Contribution

It provides novel sufficient conditions for boundedness in polymer systems, combining extended tree-operator methods with innovative partition schemes.

Findings

Resolved discrepancy between two known extensions of Dobrushin's condition

Established new conditions for uniform boundedness of correlations and free energy

Enhanced the transfer of structural information via novel partition schemes

Abstract

We present several new sufficient conditions for uniform boundedness of the reduced correlations and free energy of an abstract polymer system in a complex multidisc around zero fugacity. They resolve a discrepancy between two incomparable and previously known extensions of Dobrushin's classic condition. All conditions arise from an extension of the tree-operator approach introduced by Fern\`{a}ndez & Procacci combined with a novel family of partition schemes of the spanning subgraph complex of a cluster. The key technique is the increased transfer of structural information from the partition scheme to a tree-operator on an enhanced space.