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

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(ρ)\, \ln{S}+c_0(ρ,ξ)+\sum_{p=1}^{\infty} \frac{c_{2p}(ρ,ξ)}{S^{p}} $ with $S=M N$, $ρ=r/s$ and $ξ=M/N$. The all coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio $ξ_{eff} = \sqrtρ\;ξ$ for free and periodic boundary conditions and $ξ_{eff} = \sqrtρ\;ξ/2$ for cylindrical boundary condition and show that all finite size correction terms are invariant under transformation $ξ_{eff} \to {1}/ξ_{eff}$.