Percolation, Second Edition
List of Contents
Preface
1 What is Percolation?
- 1.1 Modelling a Random Medium
- 1.2 Why Percolation?
- 1.3 Bond Percolation
- 1.4 The Critical Phenomenon
- 1.5 The Main Questions
- 1.6 Site Percolation
- 1.7 Notes
2 Some Basic Techniques
- 2.1 Increasing Events
- 2.2 The FKG Inequality
- 2.3 The BK Inequality
- 2.4 Russo's Formula
- 2.5 Inequalities of Reliability Theory
- 2.6 Another Inequality
- 2.7 Notes
3 Critical Probabilities
- 3.1 Equalities and Inequalities
- 3.2 Strict Inequalities
- 3.3 Enhancements
- 3.4 Bond and Site Critical Probabilities
- 3.5 Notes
4 The Number of Open Clusters per Vertex
- 4.1 Definition
- 4.2 Lattice Animals and Large Deviations
- 4.3 Differentiability of $\kappa$
- 4.4 Notes
5 Exponential Decay
- 5.1 Mean Cluster Size
- 5.2 Exponential Decay of the Radius Distribution beneath $\pc$
- 5.3 Using Differential Inequalities
- 5.4 Notes
6 The Subcritical Phase
- 6.1 The Radius of an Open Cluster
- 6.2 Connectivity Functions and Correlation Length
- 6.3 Exponential Decay of the Cluster Size Distribution
- 6.4 Analyticity of $\kappa$ and $\chi$
- 6.5 Notes
7 Dynamic and Static Renormalization
- 7.1 Percolation in Slabs
- 7.2 Percolation of Blocks
- 7.3 Percolation in Half-Spaces
- 7.4 Static Renormalization
- 7.5 Notes
8 The Supercritical Phase
- 8.1 Introduction
- 8.2 Uniqueness of the Infinite Open Cluster
- 8.3 Continuity of the Percolation Probability
- 8.4 The Radius of a Finite Open Cluster
- 8.5 Truncated Connectivity Functions and Correlation Length
- 8.6 Sub-Exponential Decay of the Cluster Size Distribution
- 8.7 Differentiability of $\t$, $\chi^{\rmf}$, and $\kappa$
- 8.8 Geometry of the Infinite Open Cluster
- 8.9 Notes
9 Near the Critical Point: Scaling Theory
- 9.1 Power Laws and Critical Exponents
- 9.2 Scaling Theory
- 9.3 Renormalization
- 9.4 The Incipient Infinite Cluster
- 9.5 Notes
10 Near the Critical Point: Rigorous Results
- 10.1 Percolation on a Tree
- 10.2 Inequalities for Critical Exponents
- 10.3 Mean Field Theory
- 10.4 Notes
11 Bond Percolation in Two Dimensions
- 11.1 Introduction
- 11.2 Planar Duality
- 11.3 The Critical Probability Equals $\frac12$
- 11.4 Tail Estimates in the Supercritical Phase
- 11.5 Percolation on Subsets of the Square Lattice
- 11.6 Central Limit Theorems
- 11.7 Open Circuits in Annuli
- 11.8 Power Law Inequalities
- 11.9 Inhomogeneous Square and Triangular Lattices
- 11.10 Notes
12 Extensions of Percolation
- 12.1 Mixed Percolation on a General Lattice
- 12.2 $AB$ Percolation
- 12.3 Long-Range Percolation in One Dimension
- 12.4 Surfaces in Three Dimensions
- 12.5 Entanglement in Percolation
- 12.6 Rigidity in Percolation
- 12.7 Invasion Percolation
- 12.8 Oriented Percolation
- 12.9 First-Passage Percolation
- 12.10 Continuum Percolation
13 Percolative Systems
- 13.1 Capacitated Networks
- 13.2 Random Electrical Networks
- 13.3 Stochastic Pin-Ball
- 13.4 Fractal Percolation
- 13.5 Contact Model
- 13.6 Random-Cluster Model
Appendix I. The Infinite-Volume Limit for Percolation
Appendix II. The Subadditive Inequality
List of Notation
References
Index of Names
Subject Index