Planar dimers and Harnack curves

Abstract

In this article we study the connection between dimers and Harnack curves discovered in [15]. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the space of Harnack curves of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates. We characterize Harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their Ronkin function with given boundary conditions