Dimers, Tilings and Trees
Kenyon, Richard ; Sheffield, Scott
arXiv, 0310195 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2003-10-13
Classification:  Mathematics - Combinatorics,  Mathematical Physics,  Mathematics - Probability 
@article{0310195,
     author = {Kenyon, Richard and Sheffield, Scott},
     title = {Dimers, Tilings and Trees},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0310195}
}

Kenyon, Richard; Sheffield, Scott. Dimers, Tilings and Trees. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0310195/