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A variational principle for a non-integrable model

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Abstract

We show the existence of a variational principle for graph homomorphisms from \(\mathbb {Z}^m\) to a d-regular tree. The technique is based on a discrete Kirszbraun theorem and a concentration inequality obtained through the dynamics of the model. As another consequence of the concentration inequality we also obtain the existence of a continuum of translation-invariant ergodic gradient Gibbs measures for graph homomorphisms from \(\mathbb {Z}^m\) to a regular tree. The method is sufficiently robust such that it could be applied to other discrete models with a quite general target graphs.

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Acknowledgements

The authors want to thank Marek Biskup, Nishant Chandgotia, Filippo Colomo, Nicolas Destainville, Richard Kenyon, Michel Ledoux, Igor Pak, Laurent Saloff-Coste, Scott Sheffield, Peter Winkler and Tianyi Zheng for helpful discussions and comments. The authors also would like to express special thanks of gratitude to the anonymous referees and to Andrew Krieger. Their precise comments and questions helped a lot to further improve the manuscript.

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Correspondence to Georg Menz.

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Menz, G., Tassy, M. A variational principle for a non-integrable model. Probab. Theory Relat. Fields (2020). https://doi.org/10.1007/s00440-020-00959-w

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Keywords

  • Variational principles
  • Non-integrable models
  • Limit shapes
  • Domino tilings
  • Glauber dynamics
  • Azuma–Hoeffding
  • Gradient-Gibbs measures
  • Entropy
  • Concentration inequalities
  • Ergodic measures

Mathematics Subject Classification

  • Primary 82B20
  • 82B30
  • 82B41
  • Secondary 60J10