Exact asymptotic expansion for the resistance between the center node and a node on the cobweb network boundary

The classic problem in electrical circuit theory, first studied by Kirchhoff in 1847, is the calculation of the resistance between two arbitrary nodes in a resistor network [1]. Besides its long-standing importance in electric circuit theory, the computation of resistances is also connected to a wide range of problems as diverse as random walks [2, 3], first-passage processes [4], lattice Green’s functions [5] and classical transport in disorder media [6–8]. In 2004 Wu [9] derived a closed-form expression for the two-point resistance in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the network. Quite recently, Izmailian, Kenna and Wu [10] revisited the problem of two-point resistance and derived a new and simpler expression for the resistance between two arbitrary nodes for finite networks with resistors r and s in the two spatial directions. The new expression was then applied to the cobweb resistor network [10]. Essam and Wu [11] used one of the results [9] to derive the asymptotic expansion for the corner-tocorner resistance R(r, s) on an M ×N rectangular resistor network under free boundary conditions. This was extended by Izmailian and Huang [12] to other boundary conditions. In recent decades, the finitesize scaling and finite-size corrections in finite critical systems and their boundary effects have attracted much attention [11–30]. Of particular importance in such studies are exact results wherein the analysis can be carried out without numerical errors. In this paper we derive the exact asymptotic expansion for the resistance between the central node and a node on the boundary of the cobweb network. We show that this expansion can be written in the form