  We study random surfaces which arise as height functions of random perfect
matchings (a.k.a. dimer configurations) on an weighted, bipartite, doubly
periodic graph G embedded in the plane. We derive explicit formulas for the
surface tension and local Gibbs measure probabilities of these models. The
answers involve a certain plane algebraic curve, which is the spectral curve of
the Kasteleyn operator of the graph. For example, the surface tension is the
Legendre dual of the Ronkin function of the spectral curve. The amoeba of the
spectral curve represents the phase diagram of the dimer model. Further, we
prove that the spectral curve of a dimer model is always a real curve of
special type, namely it is a Harnack curve. This implies many qualitative and
quantitative statement about the behavior of the dimer model, such as existence
of smooth phases, decay rate of correlations, growth rate of height function
fluctuations, etc.
