Extension of Foster’s averaging formula to infinite networks with moderate growth

Abstract

An infinite resistive network (F, r) consists of an infinite connected locally finite graph F = (V ,E) and of a function r which assigns to each edge of F a positive real number called resistence of the edge. The network is energized by a finite number of external current sources represented by a 0-chain. A current in the network is defined as a 1-chain on F satisfying Kirchhoff's taws. The first rigorous study of infinite, locally finite networks is due to H. Flanders [Fll], who proved that, when the network is energized by finitely many current and voltage sources, there exists a current with finite energy flowing in the network, and that such a current is unique if it satisfies the further condition of being a limit of finite currents. An existence and uniqueness theorem that allows infinitely many sources was proved by Zemanian [Zel ], always under the assumption of finite energy. After these seminal papers many authors developed the theory of infinite networks: we refer to the bibliography in the monograph [Ze2]. In 1949, Foster [Fo] proved his celebrated formula on the averaging of the effective resistances across the edges of a finite network. Namely, if n is the cardinality of the vertex set V and R(x,y) denotes the effective resistance across the edge [x,y], then 1 R(x, y) -2 Z r(x,y~ n 1 (0) [x,y]EE