Percolation

By: Bela Bollobas, Oliver Riordan

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Hardcover | 21 September 2006

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334 Pages

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22.86 x 15.24 x 1.91

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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.

Industry Reviews

'This book contains a complete account of most of the important results in the fascinating area of percolation. Elegant and straightforward 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.' Zentralblatt MATH

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Mathematical Modelling Probability & Statistics Science Physics Materials & States of Matter Condensed Matter Physics including Liquid State & Solid State Physics Engineering & Technology

Technology in General Mechanical Engineering & Materials Materials Science

Preface	p. ix
Basic concepts and results	p. 1
Probabilistic tools	p. 36
Bond percolation on Z[superscript 2] - the Harris-Kesten Theorem	p. 50
The Russo-Seymour-Welsh method	p. 57
Harris's Theorem	p. 61
A sharp transition	p. 63
Kesten's Theorem	p. 67
Dependent percolation and exponential decay	p. 70
Sub-exponential decay	p. 76
Exponential decay and critical probabilities - theorems of Menshikov and Aizenman & Barsky	p. 78
The van den Berg-Kesten inequality and percolation	p. 78
Oriented site percolation	p. 80
Almost exponential decay of the radius - Menshikov's Theorem	p. 90
Exponential decay of the radius	p. 104
Exponential decay of the volume - the Aizenman-Newman Theorem	p. 107
Uniqueness of the infinite open cluster and critical probabilities	p. 117
Uniqueness of the infinite open cluster - the Aizenman-Kesten-Newman Theorem	p. 117
The Harris-Kesten Theorem revisited	p. 124
Site percolation on the triangular and square lattices	p. 129
Bond percolation on a lattice and its dual	p. 136
The star-delta transformation	p. 148
Estimating critical probabilities	p. 156
The substitution method	p. 156
Comparison with dependent percolation	p. 162
Oriented percolation on Z[superscript 2]	p. 167
Non-rigorous bounds	p. 175
Conformal invariance - Smirnov's Theorem	p. 178
Crossing probabilities and conformal invariance	p. 178
Smirnov's Theorem	p. 187
Critical exponents and Schramm-Loewner evolution	p. 232
Continuum percolation	p. 240
The Gilbert disc model	p. 241
Finite random geometric graphs	p. 254
Random Voronoi percolation	p. 263
Bibliography	p. 299
Index	p. 319
List of notation	p. 322
Table of Contents provided by Ingram. All Rights Reserved.

Percolation

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