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Rigidity of proper colorings of \({\mathbb {Z}}^{d}\)

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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 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).

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Notes

  1. Note that every coloring in \(\Omega ^{\text {free}}_{[n]^d}\) may be extended to a proper coloring of all of \({\mathbb {Z}}^d\), e.g., by iterated reflections (similarly to Fig. 7A).

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Acknowledgements

We thank Raimundo Briceño, Nishant Chandgotia, Ohad Feldheim and Wojciech Samotij for early discussions on proper colorings and other graph homomorphisms. We are grateful to Christian Borgs for valuable advice on the way to present the material of this paper and its companion [54]. We thank Michael Aizenman, Jeff Kahn, Eyal Lubetzky, Dana Randall, Alan Sokal, Prasad Tetali and Peter Winkler for useful discussions and encouragement. The presentation benefited significantly from the insightful comments of three anonymous referees.

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  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Ron Peled & Yinon Spinka

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Yinon Spinka

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  1. Ron Peled
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Research of both authors was supported by the Israel Science Foundation Grant 861/15 and the European Research Council starting Grant 678520 (LocalOrder). Research of Y.S. was additionally supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

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Peled, R., Spinka, Y. Rigidity of proper colorings of \({\mathbb {Z}}^{d}\). Invent. math. 232, 79–162 (2023). https://doi.org/10.1007/s00222-022-01164-3

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