On dimer models and coamoebas

We describe the relationship between dimer models on the real two-torus and coamoebas of curves in (\mathbb C^\times)^2 . We show, inter alia, that the dimer model obtained from the shell of the coamoeba is a deformation retract of the closed coamoeba 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