Dimers and M-Curves: Limit Shapes from Riemann Surfaces

TL;DR

This paper introduces a variational and integrable systems approach to determine the limit shapes of dimer models on Aztec diamond and hexagon domains, utilizing Riemann surface algebro-geometric structures.

Contribution

It develops a novel purely variational method incorporating M-curves to analyze dimer models, extending previous quasi-periodic setups and providing explicit formulas for limit shapes.

Findings

Explicit limit shape formulas for Aztec diamond and hexagon.

Numerical weights and shapes match simulations.

Method applies to dimers with gas regions.

Abstract

We present a general approach for the study of dimer model limit shape problems via variational and integrable systems techniques. In particular we deduce the limit shape of the Aztec diamond and the hexagon for quasi-periodic weights through purely variational techniques. Putting an M-curve at the center of the construction allows one to define weights and algebro-geometric structures describing the behavior of the corresponding dimer model. We extend the quasi-periodic setup of our previous paper [7] to include a diffeomorphism from the spectral data to the liquid region of the dimer. Our novel method of proof is purely variational and exploits a duality between the dimer height function and its dual magnetic tension minimizer and applies to dimers with gas regions. We apply this to the Aztec diamond and hexagon domains to obtain explicit expressions for the complex structure of the liquid region of the dimer as well as the height function and its dual. We compute the weights and the limit shapes numerically using the Schottky uniformization technique. Simulations and predicted results match completely.