The exact evaluation of the corner-to-corner resistance of an M x N resistor network: Asymptotic expansion

TL;DR

This paper derives an asymptotic expansion for the exact corner-to-corner resistance in large M x N resistor networks, providing precise formulas for resistance behavior as network size grows.

Contribution

It presents a novel asymptotic expansion for the exact resistance in resistor networks, improving understanding of their large-scale electrical properties.

Findings

Explicit asymptotic formula for R_{NxN} as N grows large

Precise constant terms in the resistance expansion

Demonstrates resistance growth logarithmically with network size

Abstract

We study the corner-to-corner resistance of an M x N resistor network with resistors r and s in the two spatial directions, and obtain an asymptotic expansion of its exact expression for large M and N. For M = N, r = s =1, our result is R_{NxN} = (4/pi) log N + 0.077318 + 0.266070/N^2 - 0.534779/N^4 + O(1/N^6).