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

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.