Extremal values of the occupancy fraction for the antiferromagnetic Ising model

Abstract

The Ising model is a mathematical model of magnetism which is frequently studied in statistical physics and computer science. For the antiferromagnetic version of the model, there is known to be a computational threshold in the complexity of sampling from the model at given magnetization on ∆-regular graphs. The value of this threshold can be determined by minimizing the occupancy fraction of the model, but prior to this paper an explicit formula was not known. This work solves the minimization problem for the majority of the relevant parameter space in the case ∆ = 3, determining the value of this threshold. Our methods also yield results on the minimization and maximization problems in other areas of the parameter space, painting a more complete picture of the occupancy fraction's behavior in 3-regular graphs.