# The $Z$-invariant massive Laplacian on isoradial graphs

3 Laboratoire de Mathématiques et Physique Théorique
LMPT - Laboratoire de Mathématiques et Physique Théorique
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 minus 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 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 $\mathcal{C}^k$ of the characteristic polynomial 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.
Type de document :
Article dans une revue
Inventiones Mathematicae, Springer Verlag, 2016, pp.doi:10.1007/s00222-016-0687-z
Domaine :

https://hal.archives-ouvertes.fr/hal-01426812
Contributeur : Cédric Boutillier <>
Soumis le : mercredi 4 janvier 2017 - 22:35:35
Dernière modification le : mardi 21 février 2017 - 16:44:01

### Identifiants

• HAL Id : hal-01426812, version 1
• ARXIV : 1504.00792

### Citation

Cédric Boutillier, Béatrice De Tilière, Kilian Raschel. The $Z$-invariant massive Laplacian on isoradial graphs. Inventiones Mathematicae, Springer Verlag, 2016, pp.doi:10.1007/s00222-016-0687-z. <hal-01426812>

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