Minimal bipartite dimers and higher genus Harnack curves
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.
Keywords
Cite
@article{arxiv.2112.12622,
title = {Minimal bipartite dimers and higher genus Harnack curves},
author = {Cédric Boutillier and David Cimasoni and Béatrice de Tilière},
journal= {arXiv preprint arXiv:2112.12622},
year = {2023}
}
Comments
v2: 67 pages, 9 figures; submitted version; v2 substantially extends v1: Lemma 19, Definition 35, Proposition 36, Lemma 41, Proposition 47 and Section 4.6 are new. v3: minor changes; accepted version, to appear in Probab. Math. Phys