The hard-core model places particles on a graph with no two neighbors occupied. The partition function counts valid configurations weighted by activity. Understanding where this function has zeros in the complex activity plane controls everything: approximation algorithms, phase transitions, sampling efficiency.
Peters, Reppekus, and Regts connect two seemingly separate properties. Correlation decay (very strong spatial mixing, VSSM) describes how fixing a distant vertex's state has vanishing influence on a target vertex's occupation probability — a local, probabilistic property. Zero-freeness of the partition function is a global, analytic property in the complex plane. They prove VSSM at a real parameter implies the partition function has no zeros nearby in the complex plane.
The bridge is a non-autonomous dynamical system of Mobius transformations. Each vertex in the graph's computation tree contributes a Mobius map, and the composition of these maps tracks how boundary conditions propagate inward. VSSM means the composed maps contract — orbits converge regardless of starting point. This contraction in the real dynamics extends to a neighborhood in the complex plane, and the extension is what keeps the partition function away from zero.
The structural point: a local mixing property (correlations decay) provides a global analytic guarantee (no zeros). The connection runs through dynamics — the same contraction that makes sampling efficient makes the partition function well-behaved. But the paper also proves the converse fails: a weaker mixing variant does not protect against zeros. The shield is specific to very strong mixing; weaker forms leave gaps.