While there is a large body of mathematical literature devoted to finite electrical
networks, infinite networks have received growing attention only in the last two
decades. The reason for this attention is probably the increasing interest for the
discrete methods in all branches of mathematics. Actually, there is strong indication
that, at least from the point of view of potential theory, infinite networks are a
discrete model of noncompact Riemann manifolds. This is true in two ways: on
one hand results in Riemann manifolds classification theory have been obtained by
discretizing the manifold and then arguing on the network thus obtained. On the
other hand, there are several results in potential theory on infinite networks which
are the discrete counterpart of results on Riemann manifolds.
From an historical point of view, the infinite network which was first investigated
by mathematicians (more than sixty years ago) is the infinite grid Zn (especially for
n < 3). The interest in this network was motivated by the study of the discretized
Laplace or Poisson equation in the plane or in the space. The operator obtained
by discretizing the laplacian is nothing else but the operator associated with the
simple random walk on Zn, the one which assigns probability l/2n of moving one
step in each direction.
As in the case of Zn, every infinite network is associated with an irreducible,
reversible Markov chain on the underlying graph. There is a discrete analogue of
the laplacian on every network, and solving the corresponding discrete Poisson's
equation is equivalent to solving Kirchhoff's equations. The converse is also true:
every denumerable, reversible and irreducible Markov chain is the Markov chain
associated with an electrical network. Thus there is an interplay between infinite
networks and Markov chains theory: electrical concepts and results are explained
in terms of probability theory and results on Markov chains are deduced from the
laws governing electrical networks.
In these notes we tried to give a unified (but by no means comprehensive)
approach to new developments in the theory of infinite networks. We start by
describing the abstract model of a network and by formulating Kirchhoff's equations.
Even though we point out the relationship with Markov chains and classify
networks into transient and recurrent (according to the type of the associated Markov
chain), chapters I and II are mainly devoted to exploring the model and to studying
the elementary properties of currents, resistances etc.
Chapter III the networks, existence theorems