TY - JOUR

T1 - Probability distribution of the shortest path on the percolation cluster, its backbone, and skeleton

AU - Porto, Markus

AU - Havlin, Shlomo

AU - Roman, H. Eduardo

AU - Bunde, Armin

PY - 1998

Y1 - 1998

N2 - We consider the mean distribution functions [Formula Presented] [Formula Presented] and [Formula Presented] 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 [Formula Presented] Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents [Formula Presented] and [Formula Presented] which determine the scaling behavior of the distribution functions in the limit [Formula Presented] i.e., [Formula Presented] [Formula Presented] and [Formula Presented] with [Formula Presented] where [Formula Presented] and [Formula Presented] are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for [Formula Presented] [Formula Presented] and [Formula Presented] are in very good agreement with our numerical results.

AB - We consider the mean distribution functions [Formula Presented] [Formula Presented] and [Formula Presented] 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 [Formula Presented] Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents [Formula Presented] and [Formula Presented] which determine the scaling behavior of the distribution functions in the limit [Formula Presented] i.e., [Formula Presented] [Formula Presented] and [Formula Presented] with [Formula Presented] where [Formula Presented] and [Formula Presented] are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for [Formula Presented] [Formula Presented] and [Formula Presented] are in very good agreement with our numerical results.

UR - http://www.scopus.com/inward/record.url?scp=0039673054&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.58.R5205

DO - 10.1103/PhysRevE.58.R5205

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AN - SCOPUS:0039673054

SN - 1063-651X

VL - 58

SP - R5205-R5208

JO - Physical Review E

JF - Physical Review E

IS - 5

ER -