Homology of Fortuin-Kasteleyn clusters of Potts models on the torus

TL;DR

This paper investigates the topological properties of Fortuin-Kasteleyn clusters in Potts models on a torus, providing probability formulas for various Q values and comparing them with numerical simulations.

Contribution

It generalizes previous percolation results to Potts models with Q=1, 2, 3, 4 and compares theoretical predictions with numerical data.

Findings

Theoretical probabilities match numerical results for Q=1, 2, 3.

Logarithmic corrections complicate the analysis for Q=4.

The study extends topological cluster analysis to a broader class of models.

Abstract

Topological properties of Fortuin-Kasteleyn clusters are studied on the torus. Namely, the probability that their topology yields a given subgroup of the first homology group of the torus is computed for Q=1, 2, 3 and 4. The expressions generalize those obtained by Pinson for percolation (Q=1). Numerical results are also presented for three tori of different moduli. They agree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement is not ruled out but what seems logarithmic corrections makes it harder to decide.