Uniform tiling with electrical resistors

TL;DR

This paper develops a method using lattice Green's functions to compute electrical resistance between any two nodes in infinite periodic resistor networks, providing explicit formulas and mapping complex lattices to simpler ones.

Contribution

It introduces a general approach based on Green's functions for calculating resistances in infinite periodic resistor networks, including explicit formulas and lattice mappings.

Findings

Resistance formulas for Kagome, diced, and decorated lattices

Mapping of complex lattices to triangular and square lattices

Method demonstrated to be efficient through several examples

Abstract

The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Green's function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagom\'e, the diced and the decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.