Rigidity of proper colorings of ℤ^d

proper q -coloring of a domain in 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 ℤ^d is replaced by ℤ^d_1×𝕋^d_2 with d_1≥ 2 , d=d_1+d_2 sufficiently high and 𝕋 a cycle of even length. The results address questions going back to Berker and Kadanoff (in J Phys A Math Gen 13(7):L259, 1980), Kotecký (in Phys Rev B 31(5):3088, 1985) and Salas and Sokal (in J Stat Phys 86(3):551–579, 1997).