A Multi-Body Dobrushin-Sokal Criterion -- Part I

TL;DR

This paper establishes a new criterion for the zero-freeness of partition functions in lattice gases with complex multi-body interactions, generalizing previous results and providing two different proofs.

Contribution

It introduces a multi-body Dobrushin-Sokal criterion for zero-freeness, extending prior work to more complex interactions and improving bounds on the Kirkwood-Salsburg operator.

Findings

Derived a sufficient condition for zero-freeness applicable to complex multi-body interactions.

Provided two proofs: an inductive approach and a hierarchy-based method.

Achieved a direct improvement of classical bounds for the Kirkwood-Salsburg operator.

Abstract

We derive a sufficient condition for zero-freeness of partition functions applicable to lattice gases with possibly complex-valued multi-body interactions. This includes the case of hard-core interactions and, in particular, generalises recent results by Galvin et al.\ (2024) and Bencs-Buys (2025) on zero-free polydiscs of hypergraph independence polynomials. We provide two proofs: the first generalises the inductive approach of Bencs and Buys; the second employs the Kirkwood-Salsburg hierarchy. Notably, the central argument of the second proof uses of a certain partition scheme for coverings and, as a by-product, we obtain a direct improvement of Gallavotti and Miracle-Sol\'e's (1968) bounds for the Kirkwood-Salsburg operator.