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

TL;DR

This paper establishes zero-free regions for the anti-ferromagnetic Potts model's partition function on bounded degree graphs, leading to improved bounds for approximation algorithms in graph coloring problems.

Contribution

It provides new zero-free regions for the Potts model on bounded degree graphs, enhancing understanding of phase transitions and algorithmic applications.

Findings

Zero-free regions for the partition function are identified for graphs with bounded degree.

For large enough k, there exists an open set in the complex plane where the partition function does not vanish.

Results lead to improved bounds on k for deterministic approximation algorithms for graph colorings.

Abstract

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. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$-colourings of graphs of small maximum degree.