Geometric Exponents of Dilute Loop Models

Abstract

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is $\beta=-2 \cos(\pi/\bar{\kappa})$ where the parameter $\bar{\kappa}$ is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges $c(\bar{\kappa})=13-6(\bar{\kappa}+\bar{\kappa}^{-1})$ and, for the minimal models of interest here, $\bar{\kappa}=p/p'$ where p and p′ are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p′)=(1,1),(2,3),(3,4),(4,5),(5,6),(5,7), and that of the red bonds for (p,p′)=(3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V→∞. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed.