Percolation and critical O($n$) loop configurations

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 $1\le n\le 2$. 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 $p$.

17 pages, 9 figures, 1 table

References in corpus (4)

• Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model
• Shape-dependent universality in percolation
• Cluster simulations of loop models on two-dimensional lattices
• Geometric properties of two-dimensional O(n) loop configurations

Cited by in corpus (8)

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• Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensional XY model
• Potts and percolation models on bowtie lattices
• Universal critical properties of the Eulerian bond-cubic model
• Conducting-angle-based percolation in the XY model
• Emergent topological ordered phase for the Ising-XY Model revealed by cluster-updating Monte-Carlo method
• Percolation of the Site Random-Cluster Model by Monte Carlo Method
• Exact critical points of the O($n$) loop model on the martini and the 3-12 lattices