Percolation, Second Edition

Draft Preface

Quite apart from the fact that percolation theory has its origin in an honest applied problem, it is a source of fascinating problems of the best kind for which a mathematician can wish: problems which are easy to state with a minimum of preparation, but whose solutions are apparently difficult and require new methods. At the same time, many of the problems are of interest to or proposed by statistical physicists and not dreamed up merely to demonstrate ingenuity.

As a mathematical subject, percolation is a child of the 1950s. Following the presentation by Hammersley and Morton (1954) of a paper on Monte Carlo methods to the Royal Statistical Society, Simon Broadbent contributed the following discussion:

Another problem of excluded volume, that of the random maze, may be defined as follows: A square (in two dimensions) or cubic (in three) lattice consists of ``cells'' at the interstices joined by ``paths'' which are either open or closed, the probability that a randomly-chosen path is open being $p$. A ``liquid'' which cannot flow upwards or a ``gas'' which flows in all directions penetrates the open paths and fills a proportion $\lambda_r(p)$ of the cells at the $r$th level. and fills a proportion $\lambda_r(p)$ of the cells at the $r$th level. The problem is to determine $\lambda_r(p)$ for a large lattice. Clearly it is a non-decreasing function of $p$ and takes the values $0$ at $p=0$ and $1$ at $p=1$. Its value in the two-dimension case is not greater than in three dimensions. It appears likely from the solution of a simplified version of the problem that as $r \to \infty$ $\lambda_r(p)$ tends strictly monotonically to $\Lambda(p)$, a unique and stable proportion of cells occupied, independent of the way the liquid or gas is introduced into the first level. No analytical solution for a general case seems to be known.

This discussion led to a fruitful partnership between Broadbent and Hammersley, and resulted in their famous paper of 1957. The subsequent publications of Hammersley initiated the mathematical study of the subject.

Much progress has been made since, and many of the open problems of the last decades have been solved. With such solutions we have seen the evolution of new techniques and questions, and the consequent knowledge has shifted the ground under percolation. The mathematics of percolation is now fairly mature, although there are major questions which remain largely unanswered. Percolation technology has emerged as a cornerstone of the theory of disordered physical systems, and the methods of this book are now being applied and extended in a variety of important settings.

The quantity of literature related to percolation seems to grow hour by hour, mostly in the physics journals. It has become difficult to get to know the subject from scratch, and one of the principal purposes of this book is to remedy this. Percolation has developed a reputation for being hard as well as important. Nevertheless, it may be interesting to note that the level of mathematical preparation required to read this book is limited to some elementary probability theory and real analysis at the undergraduate level. Readers knowing a little advanced probability theory, ergodic theory, graph theory, or mathematical physics will not be disadvantaged, but neither will their knowledge aid directly their understanding of most of the hard steps.

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 latt ice $\zz^d$. 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.

Almost all the results and arguments of this book are valid for all bond and site percolation models, subject to minor changes only; the principal exceptions are those results of Chapter 11 which make use of the self-duality of bond percolation on the square lattice. I have no especially convincing reason for my decision to study bond percolation rather than the more general case of site percolation, but was swayed in this direction by historical reasons as well as the consequential easy access to the famous exact calculation of the critical probability of bond percolation on the square lattice. In addition, unlike the case of site models, it is easy to formulate a bond model having interactions which are long-range rather than merely nearest-neighbour. Such arguments indicate the scanty importance associated with this decision.

Here are a few words about the contents of this book. In the introductory Chapter 1 we prove the existence of a critical value $\pc$ for the edge-probability $p$, marking the arrival on the scene of an infinite open cluster. The next chapter contains a general account of the three basic techniques---the FKG and BK inequalities, and Russo's formula---together with certain other useful inequalities, some drawn from reliability theory. Chapter 3 contains a brief account of numerical equalities and inequalities for critical points, together with a general method for establishing strict inequalities. This is followed in Chapter 4 by material concerning the number of open clusters per vertex. Chapters 5 and 6 are devoted to subcritical percolation (with $p<\pc$). These chapters begin with the Menshikov and Aizenman--Barsky methods for identifying the critical point, and they continue with a systematic study of the subcritical phase. Chapters 7 and 8 are devoted to supercritical percolation (with $p>\pc$). They begin with an account of dynamic renormalization, the proof that percolation in slabs characterizes the supercritical phase, and a rigorous static renormalization argument; they continue with a deeper account of this phase. Chapter 9 contains a sketch of the physical approach to the critical phenomenon (when $p=\pc$), and includes an attempt to communicate to mathematicians the spirit of scaling theory and renormalization. Rigorous results are currently limited and are summarized inChapter 10, where may be found the briefest sketch of the Hara--Slade mean field theory of critical percolation in high dimensions. Chapter 11 is devoted to percolation in two dimensions, where the technique of planar duality leads to the famous exact calculation that $\pc =\half$ for bond percolation on $\zz^2$. The book terminates with two chapters of pencil sketches of related random processes, including continuum percolation, first-passage percolation, random electrical networks, fractal percolation, and the random-cluster model.

The first edition of this book was published in 1989. The second edition differs from the first through the reorganization of certain material, and through the inclusion of fundamental new material having substantial applications in broader contexts. In particular, the present volume includes accounts of strict inequalities between critical points, the relationship between percolation in slabs and in the whole space, the Burton--Keane proof of the uniqueness of the infinite cluster, the lace expansion and mean field theory, and numerous other results of significance. A full list of references is provided, together with pointers in the notes for each chapter.

A perennial charm of percolation is the beauty and apparent simplicity of its open problems. It has not been possible to do full justice here to work currently in progress on many such problems. The big new open challenge at the time of writing is to understand the proposal that critical percolation models in two dimensions are conformally invariant. Numerical experiments support this proposal, but rigorous verification is far from complete. While a full account of conformal invariance must await a later volume, at the ends of Chapters 9 and 11 may be found lists of references and a statement of Cardy's formula.

Most of the first edition of this book was written in draft form while I was visiting Cornell University for the spring semester of 1987, a visit assisted by a grant from the Fulbright Commission. It is a pleasure to acknowledge the assistance of Rick Durrett, Michael Fisher, Harry Kesten, Roberto Schonmann, and Frank Spitzer during this period. The manuscript was revised during the spring semester of 1988, which I spent at the University of Arizona at Tucson, with financial support from the Center for the Study of Complex Systems and AFOSR contract no.\ F49620-86-C-0130. One of the principal benefits of this visit was the opportunity for unbounded conversations with David Barsky and Chuck Newman. Rosine Bonay was responsible for the cover design and index of the first edition.

In writing the second edition, I have been aided by partial financial support from the Engineering and Physical Sciences Research Council under contract GR/L15425. I am grateful to Sarah Shea-Simonds for her help in preparing the {\TeX}script of this edition, and to Alexander Holroyd for reading and commenting on parts of it.

I make special acknowledgement to John Hammersley; not only did he oversee the early life of percolation, but also his unashamed love of a good problem has been an inspiration to many.

Unstinting in his help has been Harry Kesten. He read and commented in detail on much of the manuscript of the first edition, his suggestions for improvements being so numerous as to render individual acknowledgements difficult. Without his support the job would have taken much longer and been done rather worse, if at all.