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Inventiones mathematicae

, Volume 208, Issue 1, pp 109–189 | Cite as

The Z-invariant massive Laplacian on isoradial graphs

  • Cédric Boutillier
  • Béatrice de TilièreEmail author
  • Kilian Raschel
Article

Abstract

We introduce a one-parameter family of massive Laplacian operators \((\Delta ^{m(k)})_{k\in [0,1)}\) defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of \(\Delta ^{m(k)}\), the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at \(k=0\), thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon (Invent Math 150(2):409–439, 2002) are critical. We prove that the massive Laplacian operators \((\Delta ^{m(k)})_{k\in (0,1)}\) provide a one-parameter family of Z-invariant rooted spanning forest models. When the isoradial graph is moreover \({\mathbb {Z}}^2\)-periodic, we consider the spectral curve of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus 1. We further show that every Harnack curve of genus 1 with \((z,w)\leftrightarrow (z^{-1},w^{-1})\) symmetry arises from such a massive Laplacian.

Notes

Acknowledgments

We warmly thank Erwan Brugallé for very helpful discussions on Harnack curves. We acknowledge support from the Agence Nationale de la Recherche (projet MAC2: ANR-10-BLAN-0123) and from the Région Centre-Val de Loire (projet MADACA). We thank the referee for her/his useful comments and suggestions which led us to improve the presentation of this paper. We also thank her/him for the arguments allowing to greatly simplify the proof of Proposition 6 and of Theorem 41.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Cédric Boutillier
    • 1
  • Béatrice de Tilière
    • 2
    Email author
  • Kilian Raschel
    • 3
  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité Pierre et Marie CurieParisFrance
  2. 2.Laboratoire d’Analyse et de Mathématiques AppliquéesUniversité Paris-Est CréteilCréteilFrance
  3. 3.CNRS, Laboratoire de Mathématiques et Physique ThéoriqueUniversité de ToursToursFrance

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