Dimers, tilings and trees

Generalizing results of Temperley [11], Brooks, Smith, Stone and Tutte [1] and others [10, 7] we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees of the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to “discrete analytic functions” on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting “almost periodic” tilings and harmonic functions.