  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 (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 $\ell$ 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 to
the 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 the
weak disorder regime, there is a length scale $\xi\sim a^\nu$, below which
strong disorder and critical percolation characterize the current path.
