逾渗

1 What is Percolation？1.1 Modelling a Random Medium1.2 Why Percolation？1.3 Bond Percolation1.4 The Critical Phenomenon1.5 The Main Questions1.6 Site Percolation1.7 Notes2 Some Basic Techniques2.1 Increasing Events2.2 The FKG Inequality2.3 The BK Inequality2.4 Russo's Formula2.5 Inequalities of Reliability Theory2.6 Another Inequality2.7 Notes3 Critical Probabilities3.1 Equalities and Inequalities3.2 Strict Inequalities3.3 Enhancements3.4 Bond and Site Critical Probabilities3.5 Notes4 The Number of Open Clusters per Vertex4.1 Definition4.2 Lattice Animals and Large Deviations4.3 Differentiability of K4.4 Notes5 Exponential Decay5.1 Mean Cluster Size5.2 Exponential Decay of the Radius Distribution beneath Pe5.3 Using Differential Inequalities5.4 Notes6 The Subcritical Phase6.1 The Radius of an Open Cluster6.2 Connectivity Functions and Correlation Length6.3 Exponential Decay of the Cluster Size Distribution6.4 Analyticity of K and X6.5 Notes7 Dynamic and Static Renormalization7.1 Percolation in Slabs7.2 Percolation of Blocks7.3 Percolation in Half-Spaces7.4 Static Renormalization7.5 Notes8 The Supercritical Phase8.1 Introduction8.2 Uniqueness of the Infinite Open Cluster8.3 Continuity of the Percolation Probability8.4 The Radius of a Finite Open Cluster8.5 Truncated Connectivity Functions and Correlation Length8.6 Sub-Exponential Decay of the Cluster Size Distribution8.7 Differentiability of8.8 Geometry of the Infinite Open Cluster8.9 Notes9 Near the Critical Point： Scaling Theory9.1 Power Laws and Critical Exponents9.2 Scaling Theory9.3 Renormalization9.4 The Incipient Infinite Cluster9.5 Notes10 Near the Critical Point：Rigorous Results10.1 Percolation on a Tree10.2 Inequalities for Critical Exponents10.3 Mean Field Theory10.4 Notes11 Bond Percolation in Two Dimensions12 Extensions of Percolation13 Pereolative SystemsAppendix Ⅰ The Infinite-Volume Limit for PercolationAppendix Ⅱ The Subadditive InequalityList of NotationReferencesIndex of NamesSubject Index

　　This book is about the mathematics of percolation theory，withthe emphasis upon presenting the shortest rigorous proofs of themain facts.I have made certain sacrifices in order to maximize theaccessibility of the theory，and the major one has been to restrictmyself almost entirely to the special case of bond percolation onthe cubic lattice Zd.Thus there is only little discussion of suchprocesses as continuum，mixed，inhomogeneous，long-range，first-passage，and oriented percolation.Nor have I spent much timeor space on the relationship of percolation to statisticalphysics，infinite particle systems，disordered media，reliabilitytheory，and so on.With the exception of the two final chapters，Ihave tried to stay reasonably close to core material of the sortwhich most graduate students in the area might aspire to know.Nocritical reader will agree entirely with my selection，andphysicists may sometimes feel that my intuition is crooked.