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Minimal bipartite dimers and higher genus Harnack curves

Cédric Boutillier, David Cimasoni and Béatrice de Tilière

Vol. 4 (2023), No. 1, 151–208
Abstract

This paper completes the comprehensive study of the dimer model on infinite minimal graphs with Fock’s weights (arXiv:1503.00289 (2015)) initiated in Comm. Math. Phys. (2023): the latter article dealt with the elliptic case, i.e., models whose associated spectral curve is of genus 1, while the present work applies to models of arbitrary genus. This provides a far-reaching extension of the genus 0 results of Kenyon (Invent. Math. 150:2 (2002), 409–439) and Kenyon and Okounkov (Duke Math. J. 131:3 (2006), 499–524), from isoradial graphs with critical weights to minimal graphs with weights defining an arbitrary spectral data. For any minimal graph with Fock’s weights, we give an explicit local expression for a two-parameter family of inverses of the associated Kasteleyn operator. In the periodic case, this allows us to prove local formulas for all ergodic Gibbs measures, thus providing an alternative description of the measures constructed in Ann. of Math. 163:3 (2006), 1019–1056. We also compute the corresponding slopes, exhibit an explicit parametrization of the spectral curve, identify the divisor of a vertex, and build on Ann. Sci. Éc. Norm. Supér. 46:5 (2013), 747–813 and Duke Math. J. 131:3 (2006), 499–524 to establish a correspondence between Fock’s models on periodic minimal graphs and Harnack curves endowed with a standard divisor.

Keywords
dimer model, minimal bipartite graphs, Harnack curves, Kasteleyn operator
Mathematical Subject Classification
Primary: 05C10, 30F99, 82B20
Milestones
Received: 16 March 2022
Revised: 26 October 2022
Accepted: 8 November 2022
Published: 29 March 2023
Authors
Cédric Boutillier
Sorbonne Université
CNRS
Paris
France
David Cimasoni
Section de Mathématiques
Université de Genève
Genève
Switzerland
Béatrice de Tilière
PSL University-Dauphine
CNRS
Paris
France