Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

TL;DR

This paper reviews the fractal and multifractal properties of electrical conduction in random resistor networks, highlighting analytical methods, percolation behavior, and the relation to random walks, emphasizing the scaling of conductance near the percolation threshold.

Contribution

It provides a comprehensive overview of the multifractal scaling of current distributions and their implications for electrical conductance in large random resistor networks.

Findings

Conductance vanishes at the percolation threshold from above.

Current distribution exhibits multifractal scaling at criticality.

Relation established between resistor networks and random walk phenomena.

Abstract

This article is a mini-review about electrical current flows in networks from the perspective of statistical physics. We briefly discuss analytical methods to solve the conductance of an arbitrary resistor network. We then turn to basic results related to percolation: namely, the conduction properties of a large random resistor network as the fraction of resistors is varied. We focus on how the conductance of such a network vanishes as the percolation threshold is approached from above. We also discuss the more microscopic current distribution within each resistor of a large network. At the percolation threshold, this distribution is multifractal in that all moments of this distribution have independent scaling properties. We will discuss the meaning of multifractal scaling and its implications for current flows in networks, especially the largest current in the network. Finally, we discuss the relation between resistor networks and random walks and show how the classic phenomena of recurrence and transience of random walks are simply related to the conductance of a corresponding electrical network.