Critical exponents for the homology of FortuinKasteleyn clusters on a torus
Abstract
A FortuinKasteleyn cluster on a torus is said to be of type {a,b},a,bɛZ , if it is possible to draw a curve belonging to the cluster that winds a times around the first cycle of the torus as it winds b times around the second. Even though the Q Potts models make sense only for Q integers, they can be included into a family of models parametrized by β=Q for which the FortuinKasteleyn clusters can be defined for any real βɛ(0,2] . For this family, we study the probability π({a,b}) of a given type of clusters as a function of the torus modular parameter τ=τ_{r}+iτ_{i} . We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. For example, the behavior of π({1,0}) is studied for τ_{i}→∞ . Exponents describing these behaviors are defined and related to weights h_{r,s} of the extended Kac table for r and s integers, but also halfintegers. Numerical simulations are also presented. Possible relationship with recent works and conformal loop ensembles is discussed.
 Publication:

Physical Review E
 Pub Date:
 August 2009
 DOI:
 10.1103/PhysRevE.80.021130
 arXiv:
 arXiv:0812.2925
 Bibcode:
 2009PhRvE..80b1130M
 Keywords:

 05.70.Jk;
 05.10.Ln;
 64.60.De;
 Critical point phenomena;
 Monte Carlo methods;
 Statistical mechanics of model systems;
 Condensed Matter  Statistical Mechanics
 EPrint:
 References and one figure added