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Percolation. 2nd ed. (English) Zbl 0926.60004
Grundlehren der Mathematischen Wissenschaften 321. Berlin: Springer (ISBN 978-3-540-64902-1/hbk; 978-3-642-08442-3/pbk; 978-3-662-03981-6/ebook). xiii, 444 p. (1999).
The first edition of this book was published in 1989 (Zbl 0691.60089). The second edition differs from the first one through the reorganisation of certain material and through the inclusion of several fundamental new material. For example, the state of knowledge of critical percolation greatly depends on whether the dimension \(d\) of the integer lattice is large or small. In recent years remarkable progress has been made for large \(d\) (\(d \geq 19\)). In rough terms the phase transition for large \(d\) is qualitatively similar to that of a binary tree. The method of proof is known as “lace expansion” and is presented in Chapter 10.3. Chapter 7 contains new material about the relationship between percolation in slabs and in the whole space. Another new result can be found in Chapter 8.2 where the author presents the Burton-Keane uniqueness theorem of the infinite open cluster.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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