The antiferromagnetic Potts-model spatial-mixing threshold conjecture

Let $q \geq 2$ be an integer, let $\beta \in [0,1]$, and consider the $q$-state antiferromagnetic Potts model on graphs of maximum degree $\Delta$. Define

$$\beta_c=\max\left\{0,1-\frac{q}{\Delta}\right\}.$$

Weak spatial mixing (WSM) and strong spatial mixing (SSM) mean exponential decay of the influence of boundary pinnings, with SSM allowing the pinnings to differ on an arbitrary subset of the boundary. Potts-model threshold conjecture. Both WSM and SSM hold for all graphs of maximum degree $\Delta$ if and only if $\beta \geq \beta_c$, with strict inequality when $q=\Delta$. In particular, there is a unique Gibbs measure on the infinite $\Delta$-regular tree if and only if $\beta \geq \beta_c$, again with strict inequality when $q=\Delta$. This conjectured sharp threshold would characterize both finite-graph correlation decay and uniqueness on the regular tree; the paper presents its results as improved bounds toward this widely believed statement.