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Delocalization of Uniform Graph Homomorphisms from \({\mathbb {Z}}^2\) to \({\mathbb {Z}}\)

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Abstract

Graph homomorphisms from the \({\mathbb {Z}}^d\) lattice to \({\mathbb {Z}}\) are functions on \({\mathbb {Z}}^d\) whose gradients equal one in absolute value. These functions are the height functions corresponding to proper 3-colorings of \({\mathbb {Z}}^d\) and, in two dimensions, corresponding to the 6-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered.

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Notes

  1. Functions in \(({\mathbb {Z}}^2)^{{\mathbb {Z}}^d}\) are written as \((f,g):{\mathbb {Z}}^d\rightarrow {\mathbb {Z}}^2\), where f(v) is the first value at v and g(v) is the second value at v.

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Acknowledgements

The work of NC was supported in part by the European Research Council starting grant 678520 (LocalOrder) and ISF Grant Nos. 1289/17, 1702/17, 1570/17, 2095/15 and 2919/19 and was carried out when he was a postdoctoral fellow at Tel Aviv University and the Hebrew Univeristy of Jerusalem. The work of RP was supported in part by Israel Science Foundation Grant 861/15 and the European Research Council starting Grant 678520 (LocalOrder). The work of SS was supported in part by NSF Grants DMS 1209044 and DMS 1712862. We thank Alexander Glazman and Yinon Spinka for reading our draft and suggesting various improvements. Steven M. Heilman helped prepare Fig. 1. We thank the anonymous referee for multiple suggestions for improving the clarity of the arguments.

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Authors and Affiliations

  1. Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Mumbai, India

    Nishant Chandgotia

  2. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Ron Peled

  3. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

    Scott Sheffield

  4. Department of Mathematics, Dartmouth College, Hanover, USA

    Martin Tassy

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  1. Nishant Chandgotia
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  2. Ron Peled
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Correspondence to Ron Peled.

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Communicated by H. D.-Copin.

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Chandgotia, N., Peled, R., Sheffield, S. et al. Delocalization of Uniform Graph Homomorphisms from \({\mathbb {Z}}^2\) to \({\mathbb {Z}}\). Commun. Math. Phys. 387, 621–647 (2021). https://doi.org/10.1007/s00220-021-04181-0

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