For a graph , , and complex numbers the partition function of the multivariate Potts model is defined as
where . 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 and any , there exists an open set in the complex plane that contains the interval such that for any graph of maximum degree at most and any . (Here denotes the base of the natural logarithm.) For small values of we are able to give better results.
As an application of our results we obtain improved bounds on for the existence of deterministic approximation algorithms for counting the number of proper -colourings of graphs of small maximum degree.
Cite this article
Ferenc Bencs, Ewan Davies, Viresh Patel, Guus Regts, On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), no. 3, pp. 459–489DOI 10.4171/AIHPD/108