Stochastic models with electrical analogues

Abstract

1. Introduction. It has long been known that a connexion exists between reversible random walks and electrical networks ((2), (4) pp. 303–310, (6)). In section 2 we outline this connexion. More recently Kingman (5) has exhibited an electrical analogy for a flow model. In section 3 we show that this analogy follows from the results on random walks and we extend the analogy to cover the transient behaviour of Kingman's model. In so far as this model may be regarded as a naive description of the mechanism by which electrons move in a conductor it provides a physical explanation of the connexion between random walks and electrical networks. In section 4 we describe an invasion model which was inspired by the work of Clifford and Sudbury (1) on models to represent competition between species. Adapting their method we display the relationship of this model to random walks and hence obtain an electrical analogy for another class of stochastic processes.