Shortest-Path Fractal Dimension for Percolation in Two and Three Dimensions

TL;DR

This study uses high-precision Monte Carlo simulations to accurately determine the shortest-path fractal dimension in 2D and 3D percolation, challenging previous conjectures and providing precise numerical values.

Contribution

It provides the first high-precision numerical estimates of the shortest-path fractal dimension in 2D and 3D percolation, refuting a recent conjecture in 2D.

Findings

In 2D, $ ext{d}_ ext{min} = 1.13077(2)$

In 3D, $ ext{d}_ ext{min} = 1.3756(6)$

The 2D result rules out the conjectured value $rac{217}{192}$.

Abstract

We carry out a high-precision Monte Carlo study of the shortest-path fractal dimension $\dm$ for percolation in two and three dimensions, using the Leath-Alexandrowicz method which grows a cluster from an active seed site. A variety of quantities are sampled as a function of the chemical distance, including the number of activated sites, a measure of the radius, and the survival probability. By finite-size scaling, we determine $\dm = 1.130 77(2)$ and $1.375 6(6)$ in two and three dimensions, respectively. The result in 2D rules out the recently conjectured value $\dm=217/192$ [Phys. Rev. E 81, 020102(R) (2010)].