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

TL;DR

This paper investigates the fractal geometry of two-dimensional O(n) loop models using Monte Carlo simulations and scaling analysis, confirming theoretical predictions for various n values, including noninteger ones.

Contribution

It introduces an efficient local-update Monte Carlo algorithm for O(n) models applicable to noninteger n, and provides numerical verification of theoretical fractal exponents.

Findings

Monte Carlo results match Coulomb gas predictions

Scaling exponents accurately describe fractal properties

Algorithm efficiently handles noninteger n values

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.