Trees and Matchings

In this article, Temperley’s bijection between spanning trees of a subgraph of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of an associated subgraph of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees of any planar weighted directed graph G can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related weighted planar graph H. One special case of this result is a bijection between perfect matchings of certain subgraphs of the hexagonal honeycomb lattice and directed spanning trees on associated subgraphs of a triangular lattice. Another special case gives a correspondence between perfect matchings of certain subgraphs of the “squareoctagon” lattice and directed weighted spanning trees on associated subgraphs of a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon (1997b), our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson’s algorithm allows us to quickly generate random samples of perfect matchings.