Asymptotic expansion for the resistance between two maximally separated nodes on anMbyNresistor network

Abstract

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×N rectangular network of resistors with resistors r and s in the two spatial directions. Our results is 1/s (R(M×N))(r,s) = c(ρ)ln S + c(0)(ρ,ξ) + ∑(p=1)(∞) (c(2p)(ρ,ξ))/S(p) with S = MN, 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 = square root(ρ)ξ for free and periodic boundary conditions and ξeff = square root(ρ)ξ/2 for cylindrical boundary condition and show that all finite-size correction terms are invariant under transformation ξeff→1/ξeff.