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

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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 Fernàndez & 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.