Coupling Independence Implies Zero-Freeness

For $\mathrm{\Delta }\ge 2$ and $q\ge 11\mathrm{\Delta }\mathrm{/}6$, we prove that the antiferromagnetic $q$-state Potts partition function on finite simple graphs of maximum degree at most $\mathrm{\Delta }$ has no Fisher zeros in a graph-uniform complex neighbourhood of $[0,1]$. For $q>11\mathrm{\Delta }\mathrm{/}6$, the proof establishes coupling independence throughout $[0,1]$ using a soft version of Vigoda’s flip dynamics. Our main tool is a separator-shell transfer theorem for the Potts model on induced-subgraph closed classes of graphs of maximum degree at most $\mathrm{\Delta }$, with $q\ge \mathrm{\Delta }+1$. It yields a graph-uniform zero-free neighbourhood of $[0,1]$ from Hamming coupling independence at $0$ and a uniform coupling-independence bound on each interval $[\delta ,1]$, $\delta \in (0,1]$. We also obtain zero-free Lee-Yang polydiscs around the uniform field for vertex- and edge-colour fields. For Boolean Holant problems on bounded-degree graphs whose signatures come from a fixed finite family of log-concave symmetric signatures $f$ with $f(0)>0$, such as $b$-matchings, we obtain graph-uniform zero-free polytubes around every bounded box of nonnegative activities; their union is an open zero-free neighbourhood of the nonnegative orthant. An appendix summarizes further coupling-independence inputs and the zero-free regions they yield.