ABSTRACT

This is a review of certain aspects of flows through random media. We discuss discrete media only, rather than continuous media, and we represent such media as graphical networks. We are interested in two particular types of flow – network flows and electrical flows – and both of these involve the passage of indestructible, uncreatable matter. Rather than considering the case when the constraints on the flows are deterministic, we suppose that these constraints are random variables; thus we consider network flows with random capacities and electrical networks with random resistances, and we think of these either as being “typical networks” (in the sense of operations research) or as being models for randomly–impure physical bodies. In all cases we are concerned with the asymptotic behaviour of large networks in the limit as their sizes approach infinity, and almost all of our results deal with trees, complete graphs, or subsections of crystalline lattices (such as the square lattice).