RIGIDITY OF PROPER COLORINGS OF Zd

A proper q-coloring of a domain in Z 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. Our results further characterize all periodic maximal-entropy Gibbs states for the model. Ordering also emerges in low dimensions if the lattice Z is replaced by Z1 × T2 with d1 ≥ 2, d = d1 + d2 sufficiently high and T a cycle of even length. The results address questions going back to Berker–Kadanoff (1980), Kotecký (1985) and Salas–Sokal (1997).

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