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

TL;DR

This paper investigates the probabilities and critical exponents of Fortuin-Kasteleyn clusters on a torus, extending the analysis to non-integer Q values and relating findings to conformal field theory and loop ensembles.

Contribution

It introduces a framework for analyzing FK cluster probabilities for real Q in (0,2], deriving asymptotic behaviors and exponents, and connecting these to conformal weights and recent theoretical developments.

Findings

Derived asymptotic behaviors of cluster probabilities on thin tori.

Established relationships between exponents and Kac table weights.

Presented numerical simulations supporting theoretical predictions.

Abstract

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 $\beta=\sqrt{Q}$ for which the Fortuin-Kasteleyn clusters can be defined for any real $\beta\in (0,2]$. For this family, we study the probability $\pi({\{a,b\}})$ of a given type of clusters as a function of the torus modular parameter $\tau=\tau_r+i\tau_i$. We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. For example, the behavior of $\pi(\{1,0\})$ is studied along the line $\tau_r=0$ and $\tau_i\to\infty$. 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.