Physics – Condensed Matter – Statistical Mechanics
Scientific paper
1999-04-13
Phys. Rev. E 58, 5205 (1998)
Physics
Condensed Matter
Statistical Mechanics
10 pages, 3 figures
Scientific paper
10.1103/PhysRevE.58.R5205
We consider the mean distribution functions Phi(r|l), Phi(B)(r|l), and Phi(S)(r|l), giving the probability that two sites on the incipient percolation cluster, on its backbone and on its skeleton, respectively, connected by a shortest path of length l are separated by an Euclidean distance r. Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents g1=df+dmin-d and g1B=g1S-3dmin-d, which determine the scaling behavior of the distribution functions in the limit x=r/l^(nu) much less than 1, i.e., Phi(r|l) proportional to l^(-(nu)d)x^(g1), Phi(B)(r|l) proportional to l^(-(nu)d)x^(g1B), and Phi(S)(r|l) proportional to l^(-(nu)d)x^(g1S), with nu=1/dmin, where df and dmin are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for g1, g1B, and g1S are in very good agreement with our numerical results.
Bunde Armin
Havlin Shlomo
Porto Markus
Roman Eduardo H.
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