  In this paper we study the connection between dimers and Harnack curves
discovered in math-ph/0311005. We prove that every Harnack curve arises as a
spectral curve of some dimer model. We also prove that the space of Harnack
curve 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.
