Geometric properties of two-dimensional O(n) loop configurations

Abstract

We study the fractal geometry of O($n$) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable to models with noninteger $n$ and uses local updates. Although these updates typically lead to nonlocal modifications of loop connectivities, the number of operations required per update is only of order one. The Monte Carlo algorithm is applied to the O($n$) model for several values of $n$, including noninteger ones. We thus determine scaling exponents that describe the fractal nature of O($n$) loops at criticality. The results of the numerical analysis agree with the theoretical predictions.

18 pages, 6 figures

References in corpus (1)

• Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model

Cited by in corpus (10)

• Worm Monte Carlo study of the honeycomb-lattice loop model
• Geometric Exponents, SLE and Logarithmic Minimal Models
• Percolation and critical O($n$) loop configurations
• Loop condensation in the triangular lattice quantum dimer model
• Percolation of the Site Random-Cluster Model by Monte Carlo Method
• Universal critical properties of the Eulerian bond-cubic model
• Conducting-angle-based percolation in the XY model
• Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions
• Markov chain sampling of the $O(n)$ loop models on the infinite plane
• Phase transitions in 3D Ising model with cluster weight by Monte Carlo method