ISBN: 3-540-64902-6
TITLE: Percolation
AUTHOR: Grimmett, Geoffrey
TOC:

1 What is Percolation? 1 
1.1 Modelling a Random Medium 1 
1.2 Why Percolation? 3 
1.3 Bond Percolation 9 
1.4 The Critical Phenomenon 13 
1.5 The Main Questions 20 
1.6 Site Percolation 24 
1.7 Notes 29 
2 Some Basic Techniques 32 
2.1 Increasing Events 32 
2.2 The FKG Inequality 34 
2.3 The BK Inequality 37 
2.4 Russo's Formula 41 
2.5 Inequalities of Reliability Theory 46 
2.6 Another Inequality 49 
2.7 Notes 51 
3 Critical Probabilities 53 
3.1 Equalities and Inequalities 53 
3.2 Strict Inequalities 57 
3.3 Enhancements 63 
3.4 Bond and Site Critical Probabilities 71 
3.5 Notes 75 
4 The Number of Open Clusters per Vertex 77 
4.1 Definition 77 
4.2 Lattice Animals and Large Deviations 79 
4.3 Differentiability of kappa 84 
4.4 Notes 86 
5 Exponential Decay 87 
5.1 Mean Cluster Size 87 
5.2 Exponential Decay of the Radius Distribution beneath p_c 88 
5.3 Using Differential Inequalities 102 
5.4 Notes 114 
6 The Subcritical Phase 117 
6.1 The Radius of an Open Cluster 117 
6.2 Connectivity Functions and Correlation Length 126 
6.3 Exponential Decay of the Cluster Size Distribution 132 
6.4 Analyticity of kappa and chi 142 
6.5 Notes 144 
7 Dynamic and Static Renormalization 146 
7.1 Percolation in Slabs 146 
7.2 Percolation of Blocks 148 
7.3 Percolation in Half-Spaces 162 
7.4 Static Renormalization 176 
7.5 Notes 196 
8 The Supercritical Phase 197 
8.1 Introduction 197 
8.2 Uniqueness of the Infinite Open Cluster 198 
8.3 Continuity of the Percolation Probability 202 
8.4 The Radius of a Finite Open Cluster 205 
8.5 Truncated Connectivity Functions and Correlation Length 213 
8.6 Sub-Exponential Decay of the Cluster Size Distribution 215 
8.7 Differentiability of theta, chi^f, and kappa 224 
8.8 Geometry of the Infinite Open Cluster 226 
8.9 Notes 229 
9 Near the Critical Point: Scaling Theory 232 
9.1 Power Laws and Critical Exponents 232 
9.2 Scaling Theory 239 
9.3 Renormalization 244 
9.4 The Incipient Infinite Cluster 249 
9.5 Notes 252 
10 Near the Critical Point: Rigorous Results 254 
10.1 Percolation on a Tree 254 
10.2 Inequalities for Critical Exponents 262 
10.3 Mean Field Theory 269 
10.4 Notes 278 
11 Bond Percolation in Two Dimensions 281 
11.1 Introduction 281 
11.2 Planar Duality 283 
11.3 The Critical Probability Equals  287 
11.4 Tail Estimates in the Supercritical Phase 295 
11.5 Percolation on Subsets of the Square Lattice 303 
11.6 Central Limit Theorems 309 
11.7 Open Circuits in Annuli 314 
11.8 Power Law Inequalities 324 
11.9 Inhomogeneous Square and Triangular Lattices 331 
11.10 Notes 345 
12 Extensions of Percolation 349 
12.1 Mixed Percolation on a General Lattice 349 
12.2 AB Percolation 351 
12.3 Long-Range Percolation in One Dimension 351 
12.4 Surfaces in Three Dimensions 359 
12.5 Entanglement in Percolation 362 
12.6 Rigidity in Percolation 364 
12.7 Invasion Percolation 366 
12.8 Oriented Percolation 367 
12.9 First-Passage Percolation 369 
12.10 Continuum Percolation 371 
13 Percolative Systems 378 
13.1 Capacitated Networks 378 
13.2 Random Electrical Networks 380 
13.3 Stochastic Pin-Ball 382 
13.4 Fractal Percolation 385 
13.5 Contact Model 390 
13.6 Random-Cluster Model 393 
Appendix I. The Infinite-Volume Limit for Percolation 397 
Appendix II. The Subadditive Inequality 399 
List of Notation 401 
References 404 
Index of Names 435 
Subject Index 439 
END
