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 $L\times L$ 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 $u\equiv L∕a^{\nu }$, where $\nu$ is the percolation connectedness exponent, fully determines the distribution of the current path length $\mathcal{l}$ for all values of $u$. For $u⪢1$, the behavior corresponds to the weak disorder limit and $\mathcal{l}$ scales as $\mathcal{l}\sim L$, while for $u⪡1$, the behavior corresponds to the strong disorder limit with $\mathcal{l}\sim L^{d_{\mathrm{opt}}}$, where $d_{\mathrm{opt}}=1.22\pm 0.01$ is the optimal path exponent. (b) In the weak disorder regime, there is a length scale $\xi \sim a^{\nu }$, below which strong disorder and critical percolation characterize the current path.