Rigidity of proper colorings of Z(d)

A 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(d1) x T-d2 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), Kotecky (in Phys Rev B 31(5):3088, 1985) and Salas and Sokal (in J Stat Phys 86(3):551-579, 1997).