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Limit Shapes for the Asymmetric Five Vertex Model

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Abstract

We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a weight \(r>0\). We give a variational principle for limit shapes in this setting, and show that the resulting Euler–Lagrange equation can be integrated, giving limit shapes explicitly parameterized by analytic functions.

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Acknowledgements

Research of RK is supported by NSF Grants DMS-1612668, DMS-1713033 and the Simons Foundation award 327929. JdG is support by the Australian Research Council through the ARC Centre of Excellence for Mathematical and Statistial Frontiers (ACEMS). We thank Amol Aggarwal, Hugo Duminil-Copin, Vadim Gorin, Andrei Okounkov, and István Prause for conversations and ideas about this project, and the referee for a very careful report on the paper.

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Authors and Affiliations

  1. University of Melbourne, Melbourne, Australia

    Jan de Gier

  2. Yale University, New Haven, CT, USA

    Richard Kenyon

  3. Brown University, Providence, RI, USA

    Samuel S. Watson

Authors
  1. Jan de Gier
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  2. Richard Kenyon
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  3. Samuel S. Watson
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Correspondence to Richard Kenyon.

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Communicated by H-T.Yau.

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de Gier, J., Kenyon, R. & Watson, S.S. Limit Shapes for the Asymmetric Five Vertex Model. Commun. Math. Phys. 385, 793–836 (2021). https://doi.org/10.1007/s00220-021-04126-7

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