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

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 $\xi_{eff} \to {1}/\xi_{eff}$.