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Shortest-path fractal dimension for percolation in two and three dimensions
Zongzheng Zhou1, Ji Yang, Youjin Deng
1Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China.
Abstract:
We carry out a high-precision Monte Carlo study of the shortest-path fractal dimension d(min) 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 d(min)=1.13077(2) and 1.3756(6) in two and three dimensions, respectively. The result in two dimensions rules out the recently conjectured value d(min)=217/192 [Deng et al., Phys. Rev. E 81, 020102(R) (2010)].
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