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Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus

A Fortuin-Kasteleyn cluster on a torus is said to be of type $\{a,b\}, a,b\in\mathbb Z$, if it 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 $β=\sqrt{Q}$ for which the Fortuin-Kasteleyn clusters can be defined for any real $β\in (0,2]$. 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 along the line $τ_r=0$ and $τ_i\to\infty$. Exponents describing these behaviors are defined and related to weights $h_{r,s}$ of the extended Kac table for $r,s$ integers, but also half-integers. Numerical simulations are also presented. Possible relationship with recent works and conformal loop ensembles is discussed.