Backbone three-point correlation function in the two-dimensional Potts model

Abstract: We study the three-point correlation function of the backbone in the two-dimensional $Q$-state Potts model using the Fortuin-Kasteleyn (FK) representation. The backbone is defined as the bi-connected skeleton of an FK cluster after removing all dangling ends and bridges. To circumvent the severe critical slowing down in direct Potts simulations for large $Q$, we employ large-scale Monte Carlo simulations of the O$(n)$ loop model on the hexagonal lattice, which is regarded to correspond to the Potts model with $Q=n^2$. Using a highly efficient cluster algorithm, we compute the universal three-point amplitude ratios for the backbone ($R_\text{BB}$) and FK clusters ($R_\text{FK}$). Our computed $R_\text{FK}$ exhibits excellent agreement with exact conformal field theory predictions, validating the reliability of our numerical approach. In the critical regime, we find that $R_\text{BB}$ is systematically larger than $R_\text{FK}$. Conversely, along the tricritical branch, $R_\text{BB}$ and $R_\text{FK}$ coincide within numerical accuracy, strongly suggesting that $R_\text{BB}=R_\text{FK}$ holds throughout this regime. This finding mirrors the known equality of the backbone and FK cluster fractal dimensions at tricriticality, jointly indicating that both structures share the same geometric universality.