General theory of perturbation of infinite resistor networks

TL;DR

This paper introduces an efficient method to calculate the effective resistance in infinite resistor networks with multiple bond perturbations, applicable to various physical systems and phenomena.

Contribution

We developed a novel approach to compute the Green operator for perturbed resistor networks, enabling simultaneous handling of multiple bond modifications unlike previous recursive methods.

Findings

Method successfully computes two-point resistance in complex networks.

Analytical and numerical examples demonstrate versatility and accuracy.

Insights into current distributions and broader applications are provided.

Abstract

The effective resistance between any two nodes in a perturbed resistor network is determined by removing multiple bonds from an infinite resistor lattice. We have developed an efficient method for calculating the Green operator of the Laplacian for such perturbed networks, which is directly related to the two-point resistance. Unlike the recursive techniques that remove bonds one at a time, our approach handles all bond modifications simultaneously. To demonstrate the versatility of our method, several analytical and numerical examples are presented. In addition, we computed bond current distributions to gain deeper insight into the nature of resistor perturbations. We emphasize that our method has a broad range of applications, including condensed matter physics describing the quantum mechanical effects of impurities in crystal lattices, recently emerging topoelectronics, the study of vibrations in spring networks, and problems involving random walks.