Percolation, Second Edition

List of Contents

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