×

zbMATH — the first resource for mathematics

Percolation. (English) Zbl 1118.60001
Cambridge: Cambridge University Press (ISBN 0-521-87232-4/hbk). x, 323 p. (2006).
The percolation model was introduced by S. B. Broadbent and J. M. Hammersley [Proc. Camb. Philos. Soc. 53, 629–641 (1957; Zbl 0091.13901)] for the study of random physical processes such as flows through a disordered porous medium. Although many questions in this area are easily formulated, rigorous proofs are often very difficult, and some of the fundamental classical issues remain mathematically unsolved. Thousands of papers have been devoted to the subject together with a small number of books among which the successive editions of G. Grimmett’s monography “Percolation”. New York: Springer (1989; Zbl 0691.60089) and 2nd ed. (1999; Zbl 0926.60004)] have been the standard references for a long time.
The first aim of this new book is to present classical results on bond and site percolation on a regular lattice which may or may not be oriented: edges or vertices are independently selected with the same probability \(p\) and are then called open bonds or open sites. In this setting, the fundamental question of percolation is: “For which \(p\) is there an infinite open cluster?”. The celebrated Harris-Kesten theorem asserts that the critical probability \(p_c\) for bond percolation on the square lattice is 1/2. The proof of \(p_c\geq 1/2\) goes back to T. E. Harris [Proc. Camb. Philos. Soc. 56, 13–20 (1960; Zbl 0122.36403)] and twenty years later, H. Kesten [Commun. Math. Phys. 74, 41–59 (1980; Zbl 0441.60010)] proved \(p_c\leq 1/2\), following significant progress by Russo, Seymour and Welsh. Thanks to new tools in probabilistic combinatorics a relatively simple proof can now be given for Kesten’s upper bound.
Chapter 4 describes classical results on the exponential decay of the radius and the volume of the open cluster below the critical probability. Chapter 5 presents the beautiful proof of M. R. Burton and M. Keane [Commun. Math. Phys. 121, 501–505 (1989; Zbl 0662.60113)] of uniqueness of the infinite open cluster.
Chapter 6 contains several rigorous bounds on critical probabilities in the supercritical phase. The last two chapters are devoted to recent results that have not yet appeared in book form. Chapter 7 is concerned with the conformal invariance conjecture on the limiting behaviour of certain probabilities as the lattice spacing goes to zero. Smirnov’s theorem is presented with full proof. Its consequences on the existence and values of critical exponents for site percolation on the triangular lattice are only sketched and related to the theory of Schramm-Loewner evolutions. Continuum percolation is the matter of the last chapter. The main feature here is a recent result by the authors: the critical probability for random Voronoi percolation in the plane equals 1/2. The proof is only sketched, a full proof can be found in the original 2006 paper.
This book contains a complete account of most of the important results in the fascinating area of percolation. Elegant and staightforward proofs are given with minimal background in probability or graph theory. It is self-contained, accessible to a wide readership and widely illustrated with numerous examples. It will be of considerable interest for both beginners and advanced searchers alike.

MSC:
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
82-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
PDF BibTeX XML Cite
Full Text: DOI