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**Percolation. With 76 illustrations.**
*(English)*
Zbl 0691.60089

New York etc.: Springer-Verlag. xi, 296 p. DM 98.00 (1989).

In the last years interesting and important results were obtained in percolation theory. Thus, after H. Kesten’s book [Percolation theory for mathematicians (1982; Zbl 0522.60097)] there was need for a new one, which is now at hand. Except for the last chapter, the author confines himself to the bond percolation on the d-dimensional lattice. This wise restriction makes the book accessible also to the non-expert, requiring only basic knowledge of probability theory, but leading to recent results.

The book starts with an introduction to the problems of percolation theory, followed by the presentation of basic tools. In chapter 3, the uniqueness of the critical probability is proved and in chapter 4 the number of open clusters per vertex is studied. Then the sub-critical and supercritical phases and the behavior near the critical point are examined. The bond percolation in 2 dimensions, where due to self-duality deeper results can be attained, is treated. The book closes with a survey of extensions.

The book is clearly written and the physical background well motivated. It can be recommended to mathematicians and physicists; both to those, who want to get to know this interesting theory, as to those already acquainted with it.

The book starts with an introduction to the problems of percolation theory, followed by the presentation of basic tools. In chapter 3, the uniqueness of the critical probability is proved and in chapter 4 the number of open clusters per vertex is studied. Then the sub-critical and supercritical phases and the behavior near the critical point are examined. The bond percolation in 2 dimensions, where due to self-duality deeper results can be attained, is treated. The book closes with a survey of extensions.

The book is clearly written and the physical background well motivated. It can be recommended to mathematicians and physicists; both to those, who want to get to know this interesting theory, as to those already acquainted with it.

Reviewer: M.Muermann

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

82B43 | Percolation |