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 eᵃˣ, 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 dₒₚₜ=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.