Electric currents in infinite networks

TL;DR

This survey explores the theory of electric conduction in infinite networks, extending existing work by drawing parallels with Riemann surface theory and analyzing resistance in grid networks.

Contribution

It provides a more comprehensive theoretical framework for infinite networks by linking them to the theory of open Riemann surfaces.

Findings

Analysis of resistance in d-dimensional grid networks

Extension of infinite network theory with Riemann surface parallels

Foundational results based on classical work of Flanders, Zemanian, and Thomassen

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.