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Lecture notes on particle systems and percolation. (English) Zbl 0659.60129

Wadsworth & Brooks/Cole Statistics/Probability Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. viii, 335 p. £39.95 (1988).
As the title suggests, this is a collection of lectures on selected topics in the field of interactive particle systems. It is based on six two-hour lectures given by the author and is an attempt to explain the ideas behind the proofs of the results given. As the author admits in his introduction, it is not an authoritative account of the development of the subject; it is more a selection of topics which the author believes attractive and important.
The first four chapters deal with simple growth models, the voter model, and contact processes. In chapter 5 we are led to oriented percolation; regarded as a one-dimensional phenomenon it appears before the classical two-dimensional version introduced by S. R. Broadbent and J. M. Hammersley [Proc. Cambridge Philos. Soc. 53, 629-641 (1957; Zbl 0091.139)] and from which so much has developed. Bond percolation is introduced and Kesten’s theorem that \(p=\) for the square lattice is proved. There is no mention of other lattices or the problems in higher dimensions.
Chapter 7 is about percolation properties and fractals and gives proofs of some of the assertions made by B. B. Mandelbrot in his book The fractal geometry of nature (1983; Zbl 0504.28001), Rev. ed. of “Fractals” (1977). Chapter 8 is about first passage percolation theory on the square lattice; and chapter 9 about epidemics in \({\mathbb{Z}}^ 2\). Clustering properties of the voter model are covered in chapter 10.
Within the constraints that the author admits, this is an attractive set of notes; it contains many pictures and computer simulations. However, one would have to say that for a more complete and authoritative account one would have to consult the books by H. Kesten [Percolation theory for mathematicians (1982; Zbl 0522.60097)] and T. Liggett [Interacting particle systems (1985; Zbl 0559.60078)].
Reviewer: D.J.A.Welsh

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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