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by Richard W. Kenyon, David B. Wilson
Electron. J. Combin. 7, Research Paper
http://www.math.wisc.edu/~propp/trees.ps.gz
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Abstract:
In this article, Temperley's bijection between spanning trees in the square grid and perfect matchings (also known as dimer coverings) of the square grid is generalized to the setting of general planar graphs and directed graphs, where edges may 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 perfect matchings of a related weighted planar graph H. A special case of this result gives a bijection between "lozenge " tilings of certain planar regions and directed spanning trees on associated subgraphs of a triangular lattice. In conjunction with results of Kenyon (1997b), this 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. 1.
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