Possible connection between the optimal path and flow in percolation clusters

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(ellrm optr,L) of the optimal path length ellrm opt, and find for rll L a power law decay with ellrm opt, characterized by exponent grm opt. We determine the scaling form of P(ellrm optr,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 ellrm tr of tracers inside percolation through their probability distribution P(ellrm trr,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.