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Dimers, Tilings and Trees

Richard Kenyon·Scott Sheffield
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Abstract

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others 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 on 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.

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Richard Kenyon, Scott Sheffield. 2003-10-13. Dimers, Tilings and Trees. https://arxiv.org/abs/math/0310195

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