Dimers for degenerating families of M-curves

TL;DR

This paper investigates dimer models on infinite minimal graphs associated with degenerating families of M-curves, demonstrating their consistent behavior under degeneration and their calculation as power series in deformation parameters.

Contribution

It extends the understanding of dimer models to degenerating M-curves of any genus, connecting them with Kenyon's critical models and their perturbations.

Findings

Dimer models behave consistently under degeneration of M-curves.

They can be computed as power series in deformation parameters.

The work generalizes previous models to any genus M-curves.

Abstract

We study dimer models on infinite minimal graphs with Fock's weights for degenerating families of M-curves of any genus based on works of Boutillier-Cimasoni-de Tili\`{e}re and Bobenko A.I.- Bobenko N.-Suris for a fixed M-curve. We show that these dimer models for families of M-curves behave consistently under their degeneration shrinking real circles to points, and that they can be calculated as power series in the associated deformation parameters which are regarded as the perturbation of Kenyon's critical dimer models.