Percolation and critical O($n$) loop configurations

TL;DR

This paper investigates the percolation properties of dual clusters derived from critical O(n) loop configurations on the honeycomb lattice, establishing their universal behavior and relation to the Potts model through numerical simulations.

Contribution

It introduces a novel percolation problem based on O(n) loop models, linking it to the Potts model and confirming this relation with numerical evidence.

Findings

Percolation thresholds for various n values are determined.

Universal exponents match Coulomb gas predictions.

Identifies fixed points in the renormalization flow of bond dilution.

Abstract

We study a percolation problem based on critical loop configurations of the O($n$) loop model on the honeycomb lattice. We define dual clusters as groups of sites on the dual triangular lattice that are not separated by a loop, and investigate the the bond-percolation properties of these dual clusters. The universal properties at the percolation threshold are argued to match those of Kasteleyn-Fortuin random clusters in the critical Potts model. This relation is checked numerically by means of cluster simulations of several O($n$) models in the range $1\leq n \leq 2$. The simulation results include the percolation threshold for several values of $n$, as well as the universal exponents associated with bond dilution and the size distribution of the diluted clusters at the percolation threshold. Our numerical results for the exponents are in agreement with existing Coulomb gas results for the random-cluster model, which confirms the relation between both models. We discuss the renormalization flow of the bond-dilution parameter $p$ as a function of $n$, and provide an expression that accurately describes a line of unstable fixed points as a function of $n$, corresponding with the percolation threshold. Furthermore, the renormalization scenario indicates the existence, in $p$ versus $n$ diagram, another line of fixed points at $p=1$, which is stable with respect to $p$.