On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

Understand

For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as \[ {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w, \] where $[k]:=\{1,\ldots,k\}$.

• In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs.
• In particular we show that for any $\Delta\in \mathbb{N}$ and any $k\geq e\Delta+1$, there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that ${\bf Z}(G;k,w)\neq 0$ for any $w\in U$ and any graph $G$ of maximum degree at most $\Delta$.
• (Here $e$ denotes the base of the natural logarithm.) For small values of $\Delta$ we are able to give better results.