逾渗

出版时间：2012-8  出版社：世界图书出版公司  作者：格里密特  页数：444  
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内容概要

　　This book is about the mathematics of percolation theory，with
the emphasis upon presenting the shortest rigorous proofs of the
main facts.I have made certain sacrifices in order to maximize the
accessibility of the theory，and the major one has been to restrict
myself almost entirely to the special case of bond percolation on
the cubic lattice Zd.Thus there is only little discussion of such
processes as continuum，mixed，inhomogeneous，long-range，
first-passage，and oriented percolation.Nor have I spent much time
or space on the relationship of percolation to statistical
physics，infinite particle systems，disordered media，reliability
theory，and so on.With the exception of the two final chapters，I
have tried to stay reasonably close to core material of the sort
which most graduate students in the area might aspire to know.No
critical reader will agree entirely with my selection，and
physicists may sometimes feel that my intuition is crooked.

作者简介

作者：（英国）格里密特（Geoffrey Grimmett）

书籍目录

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

章节摘录

版权页：   插图：   Sections 3.2 and 3.3.The first systematic approach to strict inequalities forordered pairs of lattices is due to Menshikov(1987a,d,e),although there existedalready some special results in the literature.The discussion and technology ofSections 3.2 and 3.3 draws heavily on Aizenman and Grimmett(1991); see alsoGrimmett(1997). Theorem(3.16)may be adapted to enhancements of site percolation(see thediscussion following the statement of the theorem).The assumption that enhancements take place at all vertices x may be relaxed; see Aizenman and Grimmett(1991). The problem of entanglements appeared first in Kantor and Hassoid(1988),who reported certain numerical conclusions.The existence of an entanglementtransition different from that of percolation was proved by Aizenman and Grimmett(1991); the strict positivity of the entanglement critical point was provedby Holroyd(1998b).The entanglement transition has been studied more systematically by Grimmett and Holroyd(1998); in particular,they have discussedcertain topological difficulties in deciding on the 'correct' definition of an infiniteentanglement and of the entanglement critical point. Related issues arise in the study of so called 'rigidity percolation',in which onestudies the existence of infinite rigid components of the open subgraph of a lattice;see Jacobs and Thorpe(1995,1996)and Holroyd(1998a).Further accounts ofentanglement and rigidity may be found in Sections 12.5 and 12.6. The 'augmented percolation' question posed after Theorem(3.16)was discussed by Chayes,Chayes,and Newman(1984)in the context of invasion perco-lation on the triangular lattice and on the covering lattice of the square lattice.Seealso Pokorny,Newman,and Meiron(1990).The question in its present form wasanswered by Aizenman and Grimmett(1991).Section 3.4.Theorem(3.28)is taken from Grimmett and Stacey(1998),wherea general theorem of this sort is presented.Earlier work on strict inequalitiesbetween bond and site critical probabilities in two dimensions may be found.

编辑推荐

《逾渗(第2版)(英文)》适合数学专业、概率论和物理数方向的老师和相关的科研人员。

用户评论 (总计5条)

•   学习了 !!!!
•   才拿到书，不错。虽然有点贵，但是很值。
•   比较专业，买的人应该不会很多。学习概率组合的人会感兴趣吧
•   Grimmett's book, Percolation, is excellent.Percolation theory began in the 50's; its mathematics is now quite mature, but the theory has recently acquired new techniques because many of the questions initially raised by percolation theory are still unanswered.Percolation technology is now a cornerstone of the theory of disordered systems, and the methods of this book are now being extended into dynamical systems theory and the life sciences. This book covers the mathematics of percolation theory, presenting the shortest rigorous proofs of the main facts. Many problems in perc...olation theory are beautiful, but some of the apparent simplicity of the subject is deceiving, because the subject is quite deep. Grimmett cuts through many of the difficulties presenting the important concepts clearly and sucinctly.The author restricts himself- for accessibility to the maximum readership-to bond percolation on a cubic lattice. Grimmett presents the core material at a graduate level for folks conversant with elementary probability theory and real analysis. Having some knowledge of ergodic theory, graph theory, and some mathematical physics helps, however. There is litle discussion of continuous, mixed, inhomogenous, long range, first passage or oriented percolation.Beginning with existance of Psubc for the edge probability p we arrive at an infinite open cluster followed by discusssion of the basic techniques of the FKC, BK inequalities and Russo's formula. Grimmett then discusses open clusters per vertex and subcritical percolation, beginning with the Aizeman-Barsky and Menshikov methods for identifying the critical point, followed by a systematic study of the subcritical phase. He then discusses supercritical percolation, including 2 dimensional percolation, continuum percolation and random processes. The author gives a full list of references. 阅读更多 ›
•   这本书非常经典，适合搞研究的。