Uniqueness for the 3-State Antiferromagnetic Potts Model on the Tree

The antiferromagnetic $q$-state Potts model is perhaps the most canonical model for which the uniqueness threshold on the tree is not yet understood, largely because of the absence of monotonicities. Jonasson established the uniqueness threshold in the zero-temperature case, which corresponds to the $q$-colourings model. In the permissive case (where the temperature is positive), the Potts model has an extra parameter $\beta\in(0,1)$, which makes the task of analysing the uniqueness threshold even harder and much less is known. In this paper, we focus on the case $q=3$ and give a detailed analysis of the Potts model on the tree by refining Jonasson's approach. In particular, we establish the uniqueness threshold on the $d$-ary tree for all values of $d\geq 2$. When $d\geq3$, we show that the 3-state antiferromagnetic Potts model has uniqueness for all $\beta\geq 1-3/(d+1)$. The case $d=2$ is critical since it relates to the 3-colourings model on the binary tree ($\beta=0$), which has non-uniqueness. Nevertheless, we show that the Potts model has uniqueness for all $\beta\in (0,1)$ on the binary tree. Both of these results are tight since it is known that uniqueness does not hold in the complementary regime. Our proof technique gives for general $q>3$ an analytical condition for proving uniqueness based on the two-step recursion on the tree, which we conjecture to be sufficient to establish the uniqueness threshold for all non-critical cases ($q\neq d+1$).