Electric currents in infinite networks

Abstract

In this survey, we present the basic facts about conduction in infinite networks. This survey is based on the work of Flanders, Zemanian, and Thomassen, who developed the theory of infinite networks from scratch. Here we show how to get a more complete theory by paralleling the well-developed theory of conduction on open Riemann surfaces. Like Flanders and Thomassen, we take as a test case for the theory the problem of determining the resistance across an edge of a d-dimensional grid of 1-ohm resistors.