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We study the current flow paths between two edges in a random resistornetwork on a $L\times L$ square lattice. Each resistor has resistance $e^{ax}$,where $x$ is a uniformly-distributed random variable and $a$ controls thebroadness of the distribution. We find (a) the scaled variable $u\equivL/a^\nu$, where $\nu$ is the percolation connectedness exponent, fullydetermines the distribution of the current path length $\ell$ for all values of$u$. For $u\gg 1$, the behavior corresponds to the weak disorder limit and$\ell$ scales as $\ell\sim L$, while for $u\ll 1$, the behavior corresponds tothe strong disorder limit with $\ell\sim L^{d_{\scriptsize opt}}$, where$d_{\scriptsize opt} = 1.22\pm0.01$ is the optimal path exponent. (b) In theweak disorder regime, there is a length scale $\xi\sim a^\nu$, below whichstrong disorder and critical percolation characterize the current path.Comment: 9 pages, 4 figure

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