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Elliptic Dimers on Minimal Graphs and Genus 1 Harnack Curves

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Abstract

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock’s elliptic weights (Fock, Inverse spectral problem for GK integrable system. arXiv e-prints arXiv:1503.00289, 2015). Specific instances of such models were studied in Boutillier et al. (Invent Math 208(1):109–189, 2017), Boutillier et al. (Probability theory and related fields, 2018) and de Tilière (Electron J Probab 26:1–86, 2021); we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of Kenyon (Invent Math 150(2):409–439, 2002) and Kenyon and Okounkov (Duke Math J 131(3):499–524, 2006) on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the \(\mathbb {Z}^2\)-periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in Kenyon et al. (Ann Math 163(3):1019–1056, 2006). We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay’s trisecant identity.

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Notes

  1. Note that in the terminology of this paper, the most natural term would be t-immersion but we chose the terminology suited to the papers cited in this section.

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Acknowledgements

This project was started when the second-named author was visiting the first and third-named authors at the LPSM, Sorbonne Université, whose hospitality is thankfully acknowledged. The first- and third-named authors are partially supported by the DIMERS project ANR-18-CE40-0033 funded by the French National Research Agency. The second-named author is partially supported by the Swiss NSF grant 200020-200400. We would like to thank Vladimir Fock for helpful discussions and inspiration, and the anonymous referee for valuable comments.

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Authors and Affiliations

  1. Laboratoire de Probabilités Statistique et Modélisation, LPSM, UMR 8001, CNRS, Sorbonne Université, 75005, Paris, France

    Cédric Boutillier

  2. CEREMADE, UMR 7534, CNRS, PSL University Paris-Dauphine, 75016, Paris, France

    David Cimasoni & Béatrice de Tilière

  3. Institut Universitaire de France, Paris, France

    Cédric Boutillier

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  1. Cédric Boutillier
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Correspondence to Cédric Boutillier.

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Boutillier, C., Cimasoni, D. & de Tilière, B. Elliptic Dimers on Minimal Graphs and Genus 1 Harnack Curves. Commun. Math. Phys. 400, 1071–1136 (2023). https://doi.org/10.1007/s00220-022-04612-6

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