Preface-- 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-- Bibliography-- Index-- List of notation.
(source: Nielsen Book Data)
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. (source: Nielsen Book Data)