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ISBN-10: 0521872324

ISBN-13: 9780521872324

Edition: 2006

Authors: Oliver Riordan, Bela Bollob�s

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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 are twofold. First to present classical results in a way that is accessible to non-specialists. Second, to describe, for the first time in a book, recent 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.
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Book details

List price: $99.99
Copyright year: 2006
Publisher: Cambridge University Press
Publication date: 9/21/2006
Binding: Hardcover
Pages: 334
Size: 6.00" wide x 9.00" long x 0.75" tall
Weight: 1.496
Language: English

Basic concepts and results
Probabilistic tools
Bond percolation on Z[superscript 2] - the Harris-Kesten Theorem
The Russo-Seymour-Welsh method
Harris's Theorem
A sharp transition
Kesten's Theorem
Dependent percolation and exponential decay
Sub-exponential decay
Exponential decay and critical probabilities - theorems of Menshikov and Aizenman & Barsky
The van den Berg-Kesten inequality and percolation
Oriented site percolation
Almost exponential decay of the radius - Menshikov's Theorem
Exponential decay of the radius
Exponential decay of the volume - the Aizenman-Newman Theorem
Uniqueness of the infinite open cluster and critical probabilities
Uniqueness of the infinite open cluster - the Aizenman-Kesten-Newman Theorem
The Harris-Kesten Theorem revisited
Site percolation on the triangular and square lattices
Bond percolation on a lattice and its dual
The star-delta transformation
Estimating critical probabilities
The substitution method
Comparison with dependent percolation
Oriented percolation on Z[superscript 2]
Non-rigorous bounds
Conformal invariance - Smirnov's Theorem
Crossing probabilities and conformal invariance
Smirnov's Theorem
Critical exponents and Schramm-Loewner evolution
Continuum percolation
The Gilbert disc model
Finite random geometric graphs
Random Voronoi percolation
List of notation