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September 2009 Loop statistics in the toroidal honeycomb dimer model
Cédric Boutillier, Béatrice de Tilière
Ann. Probab. 37(5): 1747-1777 (September 2009). DOI: 10.1214/09-AOP453


The dimer model on a graph embedded in the torus can be interpreted as a collection of random self-avoiding loops. In this paper, we consider the uniform toroidal honeycomb dimer model. We prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed, the winding number of the collection of loops converges in law to a two-dimensional discrete Gaussian distribution. This is known to physicists in more generality from their analysis of toroidal two-dimensional critical loop models and their mapping to the massless free field on the torus. This paper contains the first mathematical proof of this more general physics result in the specific case of the loop model induced by a toroidal dimer model.


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Cédric Boutillier. Béatrice de Tilière. "Loop statistics in the toroidal honeycomb dimer model." Ann. Probab. 37 (5) 1747 - 1777, September 2009.


Published: September 2009
First available in Project Euclid: 21 September 2009

zbMATH: 1179.60065
MathSciNet: MR2561433
Digital Object Identifier: 10.1214/09-AOP453

Primary: 60K35 , 82B20

Keywords: Dimers , loop ensemble , Winding number

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 5 • September 2009
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