Rigidity of proper colorings of ${\mathbb{Z}}^d$

Abstract

A proper $q$-coloring of a domain in ${\mathbb{Z}}^d$ is a function assigning one of $q$ colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper $q$-coloring uniformly at random, does the coloring typically exhibit long-range order? It has been known since the work of Dobrushin that no such ordering can arise when $q$ is large compared with $d$. We prove here that long-range order does arise for each $q$ when $d$ is sufficiently high, and further characterize all periodic maximal-entropy Gibbs states for the model. Ordering is also shown to emerge in low dimensions if the lattice ${\mathbb{Z}}^d$ is replaced by ${\mathbb{Z}}^{d_1}\times {\mathbb{T}}^{d_2}$ with $d_1\ge 2$, $d=d_1+d_2$ sufficiently high and $\mathbb{T}$ a cycle of even length. The results address questions going back to Berker--Kadanoff (1980), Kotecký (1985) and Salas--Sokal (1997).

53 pages, 7 figures. Many minor enhancements. Added figures