Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2004-11-02
Phys.Rev E, 71, 045101(R)(2005)
Physics
Condensed Matter
Disordered Systems and Neural Networks
9 pages, 4 figures
Scientific paper
10.1103/PhysRevE.71.045101
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.
Braunstein Lidia A.
Buldyrev Sergey V.
Havlin Shlomo
Lopez Eduardo
Stanley Eugene H.
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