Asymptotic expansion for the resistance between two maximum separated nodes on a $M \times N$ resistor network

TL;DR

This paper derives the exact asymptotic expansion for the electrical resistance between two maximally separated nodes in a rectangular resistor network, considering various boundary conditions and anisotropic resistances.

Contribution

It provides a new analytical asymptotic expansion formula for resistance in large resistor networks with explicit coefficients and invariance properties under aspect ratio transformations.

Findings

Derived the asymptotic resistance expansion with explicit coefficients.

Introduced the effective aspect ratio for different boundary conditions.

Showed invariance of correction terms under aspect ratio transformation.

Abstract

We analyze the exact formulae for the resistance between two arbitrary notes in a rectangular network of resistors under free, periodic and cylindrical boundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between two maximum separated nodes on an $M \times N$ rectangular network of resistors with resistors $r$ and $s$ in the two spatial directions. Our results is $ \frac{1}{s}R_{M\times N}(r,s)= c(\rho)\, \ln{S}+c_0(\rho,\xi)+\sum_{p=1}^{\infty} \frac{c_{2p}(\rho,\xi)}{S^{p}} $ with $S=M N$, $\rho=r/s$ and $\xi=M/N$. The all coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio $\xi_{eff} = \sqrt{\rho}\;\xi$ for free and periodic boundary conditions and $\xi_{eff} = \sqrt{\rho}\;\xi/2$ for cylindrical boundary condition and show that all finite size correction terms are invariant under transformation $\xi_{eff} \to {1}/\xi_{eff}$.