Minimal bipartite dimers and higher genus Harnack curves

TL;DR

This paper extends the analysis of the dimer model on minimal graphs to arbitrary genus spectral curves, providing explicit formulas for inverses, Gibbs measures, and a correspondence with Harnack curves, broadening previous genus-zero results.

Contribution

It introduces explicit local formulas for inverses of the Kasteleyn operator and establishes a link between Fock's models and Harnack curves for minimal graphs of any genus.

Findings

Explicit local formulas for Kasteleyn inverses

Descriptions of all ergodic Gibbs measures

Parametrization of spectral curves and divisors

Abstract

This paper completes the comprehensive study of the dimer model on infinite minimal graphs with Fock's weights [arXiv:1503.00289] initiated in [arXiv:2007.14699]: the latter article dealt with the elliptic case, i.e., models whose associated spectral curve is of genus one, while the present work applies to models of arbitrary genus. This provides a far-reaching extension of the genus zero results of [arXiv:math-ph/0202018, arXiv:math/0311062], 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 [arXiv:math-ph/0311005]. We also compute the corresponding slopes, exhibit an explicit parametrization of the spectral curve, identify the divisor of a vertex, and build on [arXiv:math/0311062, arXiv:1107.5588] to establish a correspondence between Fock's models on periodic minimal graphs and Harnack curves endowed with a standard divisor.