Current Flow in Random Resistor Networks: The Role of Percolation in Weak and Strong Disorder

TL;DR

This paper investigates how current flows in a random resistor network on a square lattice, revealing how the disorder strength influences the path length and connecting percolation theory with electrical transport properties.

Contribution

It introduces a unified scaling framework linking weak and strong disorder regimes in resistor networks through a key variable involving percolation exponents.

Findings

The distribution of current path length is fully determined by the scaled variable u.

In weak disorder, the path length scales linearly with system size L.

In strong disorder, the path length scales as L^{d_{opt}} with d_{opt} ≈ 1.22.

Abstract

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.