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 Ltimes 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 uequiv 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 ugg 1, the behavior corresponds to the weak disorder limit and ell scales as ellsim L, while for ull 1, the behavior corresponds to the strong disorder limit with ellsim L dscriptsize opt, where dscriptsize opt = 1.22pm0.01 is the optimal path exponent. (b) In the weak disorder regime, there is a length scale xisim a nu, below which strong disorder and critical percolation characterize the current path.