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

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 $1.3756(6)$ 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)].

5 pages, 4 figures

References in corpus (2)

• Some geometric critical exponents for percolation and the random-cluster model
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