Long-range order in discrete spin systems

We establish long-range order for discrete nearest-neighbor spin systems on $\mathbb{Z}^d$ satisfying a certain symmetry assumption, when the dimension $d$ is higher than an explicitly described threshold. The results characterize all periodic, maximal-pressure Gibbs states of the system. The results further apply in low dimensions provided that the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$ and $d=d_1+d_2$ sufficiently high, where $\mathbb{T}$ is a cycle of even length. Applications to specific systems are discussed in detail and models for which new results are provided include the antiferromagnetic Potts model, Lipschitz height functions, and the hard-core, Widom--Rowlinson and beach models and their multi-type extensions. We also establish a formula conjectured by Jenssen and Keevash for the topological pressure in the high-dimensional limit.