Current flow in random resistor networks: The role of percolation in weak and strong disorder

We study the current flow paths between two edges in a random resistor network on a L × L square lattice. Each resistor has resistance e ^ax, where x is a uniformly distributed random variable and a controls the broadness of the distribution. We find that: (a) The scaled variable u≡L/a ^ν, where ν is the percolation connectedness exponent, fully determines the distribution of the current path length ℓ for all values of u. For u ≫ 1, the behavior corresponds to the weak disorder limit and ℓ scales as ℓ∼L, while for u ≪ 1, the behavior corresponds to the strong disorder limit with ℓ∼L ^dopt, where d _opt=1.22±0.01 is the optimal path exponent. (b) In the weak disorder regime, there is a length scale ξ∼a ^ν, below which strong disorder and critical percolation characterize the current path.