Elliptic dimers on minimal graphs and genus 1 Harnack curves

TL;DR

This paper extends the analysis of elliptic dimer models from genus 0 to genus 1 on minimal graphs, providing explicit inverses, phase diagrams, and a connection to Harnack curves, thus advancing the understanding of elliptic integrable models.

Contribution

It generalizes the genus 0 results to genus 1, constructs explicit inverses of the Kasteleyn operator, and links elliptic dimers to Harnack curves, with applications to phase analysis and measure classification.

Findings

Explicit local inverses for the Kasteleyn operator in genus 1

Description of the phase diagram via correlation asymptotics

Establishment of a correspondence with Harnack curves of genus 1

Abstract

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [arXiv:1503.00289]. Specific instances of such models were studied in [arXiv:052711, arXiv:1612.09082, arXiv1801.00207]; we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of [arXiv:math-ph/0202018, arXiv:math/0311062] 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 [arXiv:math-ph/0311005] by Kenyon, Okounkov and Sheffield. 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.