Shortest-path fractal dimension for percolation in two and three dimensions.
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)].
Hefei 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.
Zongzheng
Z
Zhou
Ji
J
Yang
Youjin
Y
Deng
Robert M
RM
Ziff