Possible connection between the optimal path and flow in percolation clusters
Abstract
We study the behavior of the optimal path between two sites separated by a distance r on a d -dimensional lattice of linear size L with weight assigned to each site. We focus on the strong disorder limit, i.e., when the weight of a single site dominates the sum of the weights along each path. We calculate the probability distribution P(ℓopt∣r,L) of the optimal path length ℓopt , and find for r≪L a power-law decay with ℓopt , characterized by exponent gopt . We determine the scaling form of P(ℓopt∣r,L) in two- and three-dimensional lattices. To test the conjecture that the optimal paths in strong disorder and flow in percolation clusters belong to the same universality class, we study the tracer path length ℓtr of tracers inside percolation through their probability distribution P(ℓtr∣r,L) . We find that, because the optimal path is not constrained to belong to a percolation cluster, the two problems are different. However, by constraining the optimal paths to remain inside the percolation clusters in analogy to tracers in percolation, the two problems exhibit similar scaling properties.
- Publication:
-
Physical Review E
- Pub Date:
- November 2005
- DOI:
- 10.1103/PhysRevE.72.056131
- arXiv:
- arXiv:cond-mat/0510127
- Bibcode:
- 2005PhRvE..72e6131L
- Keywords:
-
- 64.60.Ak;
- 47.55.Mh;
- 05.60.Cd;
- Renormalization-group fractal and percolation studies of phase transitions;
- Classical transport;
- Condensed Matter - Disordered Systems and Neural Networks;
- Condensed Matter - Statistical Mechanics
- E-Print:
- Accepted for publication to Physical Review E. 17 Pages, 6 Figures, 1 Table