General theory of perturbation of infinite resistor networks

TL;DR

This work tackles the problem of computing two-point resistances in infinite resistor networks perturbed by multiple bond modifications. It introduces a non-recursive Green-operator method based on the Woodbury identity, yielding a compact, finite-dimensional correction to the unperturbed Green operator and enabling direct calculation of $R(r_i,r_j)$ via $R(r_i,r_j)=R_0(r_i,r_j) + (U^{(i)}-U^{(j)})\mathbf B^{-1}(V^{(i)}-V^{(j)})$. The authors derive explicit formulas for the perturbed Green operator, demonstrate the approach on square and triangular lattices with various topological defects (islands, lakes, and periodic patterns), and provide numerical results for complex perturbations, including obstacle-like shapes and text-inspired patterns. The method scales with the number of removed bonds and remains numerically stable, offering straightforward generalization to finite networks, other tilings, and RLC circuits, with broad relevance to condensed-matter physics, topoelectronics, vibrations, and stochastic processes. Overall, the paper delivers a powerful, versatile framework for analyzing lattice perturbations and their impact on transport properties.

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.

Figures (16)

• Figure 1: In an infinite square resistor network $N=84$ bonds are removed to form a string of JPA 2025. The red dot is the origin of the coordinate system. The resistance is calculated between sites $\mathbf{r}_i =\left(2,8\right)$ and $\mathbf{r}_j=\left(21,8\right)$ (blue dots).
• Figure 2: (a) Disconnected island. (b) A lake created by four single-site islands (four isolated sites). (c) Extended island and two single-site islands (two isolated sites). (d) A bridge (magenta line) connects the island to one site of the perimeter of the infinite lattice. (e) Introducing dangling bonds (magenta lines) that pass through the four isolated sites. (f) A bridge connects the extended island, and dangling bonds pass through the two isolated sites. The resistance is calculated between sites $\mathbf{r}_i$ and $\mathbf{r}_j$ (blue dots). The red dot is the origin of the coordinate system.
• Figure 3: (a) Perturbation of an infinite square resistor network by removing four bonds. (b) The bond vectors (magenta arrows) are $\mathbf{b}_1, \mathbf{b}_2$, $\mathbf{b}_3$ and $\mathbf{b}_4$. The resistance is calculated between sites $\mathbf{r}_i$ and $\mathbf{r}_j$ (blue dots). The red dot is the origin of the coordinate system.
• Figure 4: Bond current distribution of the perturbed lattice shown in figure \ref{['4bonds_sq_A:fig']}(a) for (a) $\mathbf{r}_i =\left(1,0\right)$ and $\mathbf{r}_j =\left(1,1\right)$, and (b) $\mathbf{r}_i =\left(1,0\right)$ and $\mathbf{r}_j =\left(0,1\right)$ (blue dots). The red dot is the origin of the coordinate system. The width of the arrow lines is proportional to the current value.
• Figure 5: (a) Lake made by removing seven resistors. (b, c) Two different configurations when only five resistors are removed. The dangling bonds are represented by magenta lines. The resistance is calculated between sites $\mathbf{r}_i$ and $\mathbf{r}_j$ (blue dots). The red dot is the origin of the coordinate system.