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

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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 × N rectangular network of resistors with resistors r and s in the two spatial directions. Our results is 1/sRₘ× ₙ(r,s)= c(ρ) S+c₀(ρ,ξ)+∑ₚ₌₁∞ c₂ₚ(ρ,ξ)Sᵖ 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 ξeff = √ρξ for free and periodic boundary conditions and ξeff = √ρξ/2 for cylindrical boundary condition and show that all finite size correction terms are invariant under transformation ξeff → 1/ξeff.