In this last chapter we shall survey a number of instances where infinite electrical networks are useful models of physical phenomena, or serve as analogs in some other mathematical disciplines, or are realizations of certain abstract entities. We shall simply describe those applications without presenting a detailed exposition. To do the latter would carry us too far afield into quite a variety of subjects. However, we do provide references to the literature wherein the described applications can be examined more closely.

Several examples are presented in Sections 8.1 and 8.2 that demonstrate how the theory of infinite electrical networks is helpful for finding numerical solutions of some partial differential equations when the phenomenon being studied extends over an infinite region. The basic analytical tool is an operator version of Norton's representation, which is appropriate for an infinite grid that is being observed along a boundary. In effect, the infinite grid is replaced by a set of terminating resistors and possibly equivalent sources connected to the boundary nodes. In this way, the infinite domain of the original problem can be reduced to a finite one – at least so far as one of the spatial dimensions is concerned. This can save computer time and memory-storage requirements.

In Section 8.3 we describe two classical problems in the theory of random walks on infinite graphs and state how infinite-electrical-network theory solves those problems. Indeed, resistive networks are analogs for random walks.

In 1971 Harley Flanders [51] opened a door by showing how a unique, finite-power, voltage-current regime could be shown to exist in an infinite resistive network whose graph need not have a regular pattern. To be sure, infinite networks had been examined at least intermittently from the earliest days of circuit theory, but those prior works were restricted to simple networks of various sorts, such as ladders and grids. For example, infinite uniform ladder networks were analyzed in [31], [73], and [139], works that appeared 70 to 80 years ago. Early examinations of uniform grids and the discrete harmonic operators they generate can be found in [35], [44], [45], [47], [52], [54], [65], [84], [126], [135], [143].

Flanders' theorem, an exposition of which starts this chapter, is restricted to locally finite networks with a finite number of sources. Another tacit assumption in his theory is that only open-circuits appear at the infinite extremities of the network. The removal of these restrictions, other extensions, and a variety of ramifications [158], [159], [163], [177] comprise the rest of this chapter. However, the assumptions that the network consists only of linear resistors and independent sources and is in a finite-power regime are maintained throughout this chapter.

Actually, finite-power theories for nonlinear networks are now available, and they apply just as well to linear networks as special cases. One is due to Dolezal [40], [41], and the other to DeMichelle and Soardi [37].

We have already seen that the idea of “connections at infinity” can be encompassed within electrical network theory through the invention of 1-nodes. A consequence is the genesis of transfinitely connected graphs, that is, graphs having pairs of nodes that are connected through transfinite paths but not through finite ones. An example of this is provided by Figure 3.4; there is no finite path connecting a node of branch a to a node of branch α, but there are 1-paths that do so. The maximal, finitely connected subnetworks of that figure are the 0-sections of the 1-graph, and the 1-nodes described in Example 3.2-5 connect those infinitely many 0-sections into a 1-graph.

There is an incipient inductive process arising here. Just as 0-nodes connect branches together to produce a 0-graph, so too do 1-nodes connect 0-sections together to produce 1-graphs. The purpose of the present chapter is to pursue this induction. While doing so, we will discover an infinite hierarchy of transfinite graphs. It turns out that most of the electrical network theory discussed so far can be transferred to such graphs to obtain an infinite hierarchy of transfinite networks. Thus, an electrical parameter in one branch can affect the voltage-current pair of another branch, even when every path connecting the two branches must pass through an “infinity of infinite extremities.”

To think about this in another way, let each 0-section of Figure 3.4 be replaced by replicates of the entire 1-graph of that figure.

The preceding chapters were focused on the mathematical foundations of infinite electrical networks, existence and uniqueness theorems being their principal result. Generality was a concomitant aim of those discussions. For the remaining chapters, we shift our attention to particular kinds of networks (namely, the infinite cascades and grids) that are more closely related to physical phenomena. Two examples of this significance were given in Section 1.7, and more will be discussed in Chapter 8. Our proofs will now be constructive, and consequently methods for finding voltage-current regimes will be encompassed. Moreover, various properties of voltage-current regimes will examined.

We must now be specific about any network we hope to analyze. In particular, its graph and element values need to be stipulated everywhere. An easy way of doing this is to impose some regularity upon the network. Most of this chapter (Sections 6.1 to 6.8) is devoted to the simplest of such regularities, the periodic two-times chainlike structures. They are 0-networks appearing in two forms. One form will be called a one-ended grounded cascade and is illustrated in Figure 6.1; it has an infinite node as one of its spines, and a one-ended 0-path as the other spine. The other form will be called a one-ended ungrounded cascade and is shown in Figure 6.2; in this case, both spines are one-ended 0-paths. The third possibility of both spines being infinite nodes is a degenerate case and will not be discussed.