Asymptotic expansion for the resistance between two maximally separated nodes on an M by N resistor network

We analyze the exact formulas 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 maximally separated nodes on an M x N rectangular network of resistors with resistors r and s in the two spatial directions. Our results is 1/s R-M x N(r,s) = c(rho)ln S + c(0)(rho,xi) + Sigma(infinity)(p=1)c(2p)(rho,xi)/S-p with S = MN, 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) = root rho xi for free and periodic boundary conditions and xi(eff) = root rho xi/2 for cylindrical boundary condition and show that all finite-size correction terms are invariant under transformation xi(eff)-> 1/xi(eff).