Rigidity of proper colorings of Zd

Ron Peled*, Yinon Spinka

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A proper q-coloring of a domain in Zd 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 Zd is replaced by Zd1×Td2 with d1≥ 2 , d= d1+ d2 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).

Original languageEnglish
Pages (from-to)79-162
Number of pages84
JournalInventiones Mathematicae
Issue number1
StatePublished - Apr 2023


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