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Percolation theory was initiated some fifty years ago as a mathematical framework for the study of random physical processes such as flow through a disordered porous medium. It has proved to be a remarkably rich theory, with applications beyond natural phenomena to topics such as network modelling. The aims of this book, first published in 2006, are twofold. First to present classical results in a way that is accessible to non-specialists. Second, to describe results of Smirnov in conformal invariance, and outline the proof that the critical probability for random Voronoi percolation in the plane is 1/2. Throughout, the presentation is streamlined, with elegant and straightforward proofs requiring minimal background in probability and graph theory. Numerous examples illustrate the important concepts and enrich the arguments. All-in-all, it will be an essential purchase for mathematicians, physicists, electrical engineers and computer scientists working in this exciting area.Read more
- Systematically presents probabilistic and graph theory tools for percolation, and that describes Smirnov's results in conformal invariance
- Streamlined presentation requiring only minimal background in probability and graph theory
- Numerous examples illustrate the important concepts and enrich the arguments
Reviews & endorsements
"... A useful addition for its focus on the recent advances in two-dimensional percolation..."
Rongfeng Sun, Mathematical Reviews
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- Date Published: September 2006
- format: Hardback
- isbn: 9780521872324
- length: 334 pages
- dimensions: 235 x 161 x 20 mm
- weight: 0.66kg
- contains: 112 b/w illus.
- availability: Available
Table of Contents
1. Basic concepts
2. Probabilistic tools
3. Percolation on Z2 - the Harris-Kesten theorem
4. Exponential decay and critical probabilities - theorems of Menshikov and Aizenman & Barsky
5. Uniqueness of the infinite open cluster and critical probabilities
6. Estimating critical probabilities
7. Conformal invariance - Smirnov's theorem
8. Continuum percolation
List of notation.
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