Geometric Exponents of Dilute Logarithmic Minimal Models

TL;DR

This paper measures the fractal dimensions of various geometric features in dilute logarithmic minimal models using Monte Carlo simulations, confirming theoretical predictions with high precision and proposing computational improvements.

Contribution

It provides the first extensive numerical verification of geometric exponents in dilute minimal models and introduces enhanced Monte Carlo techniques for these simulations.

Findings

Measured fractal dimensions match theoretical predictions within three to four digits.

Proposed Monte Carlo upgrades significantly improve simulation speed.

Discrepancies with theory are explained by extrapolation challenges.

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 = -2cos(pi/barkappa}) where the parameter barkappa is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges c = 13 - 6(barkappa + barkappa^{-1}) and, for the minimal models of interest here, barkappa = 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 -> infinity. 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.