Synopses & Reviews
Account of modern percolation theory and its ramifications.
Review
"... A useful addition for its focus on the recent advances in two-dimensional percolation..."
Rongfeng Sun, Mathematical Reviews
Synopsis
Percolation theory was initiated as a mathematical framework for studying random physical processes. It has proved to be remarkably rich, with applications beyond natural phenomena to topics such as network modelling. The authors' first aim is to present classical results in a way accessible to non-specialists. Second, they describe recent results of Smirnov in conformal invariance. Throughout, the presentation is streamlined, with elegant and straightforward proofs requiring minimal background. Numerous examples illustrate important concepts and enrich the arguments. All-in-all, it is essential reading for those working in this exciting area.
Synopsis
The authors' main aims are first to present classical results in a way that's accessible to non-specialists. Second, to describe, for the first time in a book, recent results of Smirnov in conformal invariance. It is essential reading for all working in this exciting area.
About the Author
Béla Bollobás is a Senior Research Fellow at Trinity College, Cambridge and is the Jabie Hardin Chair of Excellence in Combinatorics at the University of Memphis. He has held visiting positions from Seattle to Singapore, from Brazil to Zurich. This is his eleventh book.Oliver Riordan is a Royal Society Research Fellow in the University of Cambridge and Research Fellow of Trinity College, Cambridge. In May 2000 he shared the prize for solving Christopher Monckton's Eternity Puzzle.
Table of Contents
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.