Dimers for degenerating families of M-curves

TL;DR

This work extends integrable dimer models with Fock weights from fixed $M$-curves to degenerating families of $M$-curves of arbitrary genus by employing Abelian-differential variation formulas and Schottky uniformization. It shows that dimer weights, kernels, and inverses behave coherently under the degeneration that shrinks real ovals to points, and that the degenerating models admit computable power-series expansions in the deformation parameters, interpreted as perturbations of Kenyon's critical dimer model. The results imply that key dimer-theoretic objects (Amoeba maps, Ronkin functions, surface tensions, height functions) should vary nicely with degeneration since they are expressed via Abelian functions. Collectively, these findings provide a robust framework for understanding dimer models on singular limits of $M$-curves and their convergence to normalized models, with explicit perturbative tools for calculation.

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 Tilière 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.