An introduction to the dimer model

A perfect matching of a graph is a subset of edges which covers every vertex exactly once, that is, for every vertex there is exactly one edge in the set with that vertex as endpoint. The dimer model is the study of the set of perfect matchings of a (possibly infinite) graph. The most well-known example is when the graph is Z, for which perfect matchings are equivalent (via a simple duality) to domino tilings, that is, tilings of the plane with 2× 1 and 1× 2 rectangles. In the first three sections we study domino tilings of the plane and of finite polygonal regions, or equivalently, perfect matchings on Z and subgraphs of Z. In the last two sections we study the FK-percolation model and the dimer model on a more general family of planar graphs.