目录 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 K 4.4 Notes 5 Exponential Decay 5.1 Mean Cluster Size 5.2 Exponential Decay of the Radius Distribution beneath Pe 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 K and X 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 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 12 Extensions of Percolation 13 Pereolative Systems Appendix Ⅰ The Infinite-Volume Limit for Percolation Appendix Ⅱ The Subadditive Inequality List of Notation References Index of Names Subject Index
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