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

12/18/2018

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by   Ferenc Bencs, et al.

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For a graph G=(V,E), k∈N, and a complex number w the partition function of the univariate Potts model is defined as Z(G;k,w):=∑_ϕ:V→ [k]∏_uv∈ E ϕ(u)=ϕ(v)w. 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 Δ∈N and any k≥ 3.02 Δ+1, there exists an open set U in the complex plain that contains the interval [0,1) such that Z(G;k,w)≠ 0 for any w∈ U and any graph G of maximum degree at most Δ. For small values of Δ 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.