Loop statistics in the toroidal honeycomb dimer model

Details Loop statistics in the toroidal honeycomb dimer model Loop statistics in the toroidal honeycomb dimer model

2006-08-24

arxiv.org/abs/math/0608600v2

Annals of Probability 2009, Vol. 37, No. 5, 1747-1777

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP453 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

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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