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Random walks and electric networks. (English) Zbl 0583.60065
The Carus Mathematical Monographs, 22. Washington, D. C.: The Mathematical Association of America. Distr. by John Wiley & Sons, New York etc. XIII, 159 p. £22.00 (1984).
This book is an instructive example of the interplay between physical and mathematical concepts; it demonstrates how ideas from one scientific area can throw light on another.
Part I of the book surveys random walks in one and two dimensions, as well as on more general networks. It concludes with an account of Rayleigh’s monotonicity law and its probabilistic explanation: this law states that if the resistances of an electric circuit are increased (decreased), the effective resistance between any two points can only increase (decrease).
Part II deals with random walks on infinite networks and leads to Pólya’s theorem that a simple random walk on a d-dimensional lattice is recurrent for \(d=1,2\), but is transient for \(d>2\). This is proved by using Rayleigh’s short-cut method, to derive extensions of Pólya’s theorem.
The authors have written a delightful monograph which uses relatively elementary methods to tackle an important problem in the theory of random walks: it is a welcome addition to the literature in the field.
Reviewer: J.Gani

60G50 Sums of independent random variables; random walks
60-02 Research exposition (monographs, survey articles) pertaining to probability theory