The partition function of the hard-core model on a graph encodes every equilibrium property of the system. Locating its complex zeros tells you where phase transitions occur. Proving there are no zeros in a region guarantees analyticity—and with it, efficient computation.

Peters, Reppekus, and Regts introduce “very strong spatial mixing” (VSSM), a condition requiring that the influence of distant boundary conditions decays uniformly across all induced subgraphs. They prove VSSM implies zero-freeness: no complex zeros of the partition function in the corresponding region, for all graphs in the family.

The surprise is what happens when you relax the condition slightly. A closely related variant—nearly identical in formulation—does not guarantee zero-freeness. The boundary between the condition that works and the one that doesn’t is, in the authors’ analysis, razor-thin.

The proof converts the problem into analyzing a non-autonomous dynamical system of Möbius transformations, where VSSM corresponds to strict contractivity of the composed maps. The almost-identical variant fails because it permits sequences of maps that contract on average but not uniformly—and a single non-contracting step can drive the composition through the zero.

This matters because it reveals that the standard hierarchy of mixing conditions (weak → strong → very strong) has a gap at the top that is functionally infinite: everything below VSSM is compatible with zeros, while VSSM itself excludes them completely. There is no intermediate regime. The transition from “zeros possible” to “zeros impossible” is a discontinuity in the space of mixing conditions. The right version of a concept can be separated from useless variants by a subtlety invisible at first glance—and that subtlety is load-bearing.