On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs
Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 3, pp. 459-489.
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For a graph G=(V,E), k∈ℕ, and complex numbers w=(w e ) e∈E the partition function of the multivariate Potts model is defined as

$$

where [k]:={1,...,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 Δ∈ℕ and any k≥eΔ+1, there exists an open set U in the complex plane that contains the interval [0,1) such that 𝐙(G;k,w)≠0 for any graph G=(V,E) of maximum degree at most Δ and any w∈U E . (Here e denotes the base of the natural logarithm.) 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.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/108
Classification : 05-XX, 82-XX
Mots-clés : Anti-ferromagnetic Potts model, counting proper colourings, partition function, approximation algorithm, complex zeros 
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     title = {On zero-free regions for the anti-ferromagnetic {Potts} model on bounded-degree graphs},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {459--489},
     volume = {8},
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     doi = {10.4171/aihpd/108},
     mrnumber = {4321222},
     zbl = {1479.82007},
     language = {en},
     url = {http://archive.numdam.org/articles/10.4171/aihpd/108/}
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Bencs, Ferenc; Davies, Ewan; Patel, Viresh; Regts, Guus. On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs. Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 3, pp. 459-489. doi : 10.4171/aihpd/108. http://archive.numdam.org/articles/10.4171/aihpd/108/

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