On dimer models and coamoebas

TL;DR

This paper explores the connection between dimer models on a torus and coamoebas of algebraic curves, revealing conditions under which the dimer model is a deformation retract of the coamoeba and analyzing the maximality of complement components.

Contribution

It establishes a precise relationship between dimer models and coamoebas, including conditions for deformation retracts and maximal complement components.

Findings

Dimer model from coamoeba shell is a deformation retract iff the complement components are maximal.

Closed coamoeba of a characteristic polynomial generally does not have maximal complement components.

The paper clarifies the topological relationship between dimer models and coamoebas.

Abstract

We describe the relationship between dimer models on the real two-torus and coamoebas of curves in (\CC^\times)^2. We show, inter alia, that the dimer model obtained from the shell of the coamoeba is a deformation retract of the closed coaomeba if and only if the number of connected components of the complement of the closed coamoeba is maximal. Furthermore, we show that in general the closed coamoeba of the characteristic polynomial of a dimer model does not have the maximal number of components of its complement.