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\line{\bigss Geoffrey Grimmett\hfil}
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\line{\extrabigbss Percolation\hfil}
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\line{\ss Second Edition\hfil}
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\line{\ss With 121 Illustrations\hfil}
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{\baselineskip=16pt \fourteenhel
\line{Springer-Verlag\hfil}
\line{New York Berlin Heidelberg\hfil}
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{\parskip=0pt \parindent=0pt
Geoffrey Grimmett

Statistical Laboratory

University of Cambridge

16 Mill Lane

Cambridge CB2 1SB

United Kingdom

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Mathematics Subject Classification (1991): 60K35, 82B43
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\pageno=-7
\app {}. Preface to the Second Edition.
%\flushpar
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:

\smallskip
{\parindent=18pt\narrower\flushpar\llap{\lq\lq}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.
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.\rq\rq 

}
\smallskip
\flushpar

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 lattice
$\zz^d$. 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 in 
Chapter 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.
\vskip1cm
\line{\hfill G.\ R.\ G.}
\line{\hfill Cambridge}
\line{\hfill October 1998}

\vfill\eject

\app {}. Contents.

\chaptit{1}{What is Percolation?}{1}
\secttit{1.1}{Modelling a Random Medium}{1}
\secttit{1.2}{Why Percolation?}{3}
\secttit{1.3}{Bond Percolation}{9}
\secttit{1.4}{The Critical Phenomenon}{12}
\secttit{1.5}{The Main Questions}{20}
\secttit{1.6}{Site Percolation}{24}
\secttit{1.7}{Notes}{29}

\chaptit{2}{Some Basic Techniques}{32}
\secttit{2.1}{Increasing Events}{32}
\secttit{2.2}{The FKG Inequality}{33}
\secttit{2.3}{The BK Inequality}{36}
\secttit{2.4}{Russo's Formula}{41}
\secttit{2.5}{Inequalities of Reliability Theory}{46}
\secttit{2.6}{Another Inequality}{49}
\secttit{2.7}{Notes}{51}

\chaptit{3}{Critical Probabilities}{53}
\secttit{3.1}{Equalities and Inequalities}{53}
\secttit{3.2}{Strict Inequalities}{57}
\secttit{3.3}{Enhancements}{63}
\secttit{3.4}{Bond and Site Critical Probabilities}{71}
\secttit{3.5}{Notes}{75}

\chaptit{4}{The Number of Open Clusters per Vertex}{77}
\secttit{4.1}{Definition}{77}
\secttit{4.2}{Lattice Animals and Large Deviations}{79}
\secttit{4.3}{Differentiability of $\kappa$}{84}
\secttit{4.4}{Notes}{86}

\chaptit{5}{Exponential Decay}{87}
\secttit{5.1}{Mean Cluster Size}{87}
\secttit{5.2}{Exponential Decay of the Radius Distribution beneath
$\pc$}{88}
\secttit{5.3}{Using Differential Inequalities}{102}
\secttit{5.4}{Notes}{114}

\chaptit{6}{The Subcritical Phase}{117}
\secttit{6.1}{The Radius of an Open Cluster}{117}
\secttit{6.2}{Connectivity Functions and Correlation Length}{126}
\secttit{6.3}{Exponential Decay of the Cluster Size Distribution}{132}
\secttit{6.4}{Analyticity of $\kappa$ and $\chi$}{142}
\secttit{6.5}{Notes}{144}

\chaptit{7}{Dynamic and Static Renormalization}{146}
\secttit{7.1}{Percolation in Slabs}{146}
\secttit{7.2}{Percolation of Blocks}{148}
\secttit{7.3}{Percolation in Half-Spaces}{161}
\secttit{7.4}{Static Renormalization}{176}
\secttit{7.5}{Notes}{196}

\chaptit{8}{The Supercritical Phase}{197}
\secttit{8.1}{Introduction}{197}
\secttit{8.2}{Uniqueness of the Infinite Open Cluster}{198}
\secttit{8.3}{Continuity of the Percolation Probability}{202}
\secttit{8.4}{The Radius of a Finite Open Cluster}{205}
\secttit{8.5}{Truncated Connectivity Functions and Correlation
Length}{213}
\secttit{8.6}{Sub-Exponential Decay of the Cluster Size
Distribution}{215}
\secttit{8.7}{Differentiability of $\t$, $\chi^{\rmf}$, and $\kappa$}{224}
\secttit{8.8}{Geometry of the Infinite Open Cluster}{226}
\secttit{8.9}{Notes}{229}

\chaptit{9}{Near the Critical Point: Scaling Theory}{232}
\secttit{9.1}{Power Laws and Critical Exponents}{232}
\secttit{9.2}{Scaling Theory}{239}
\secttit{9.3}{Renormalization}{244}
\secttit{9.4}{The Incipient Infinite Cluster}{249}
\secttit{9.5}{Notes}{252}

\chaptit{10}{Near the Critical Point: Rigorous Results}{254}
\secttit{10.1}{Percolation on a Tree}{254}
\secttit{10.2}{Inequalities for Critical Exponents}{262}
\secttit{10.3}{Mean Field Theory}{269}
\secttit{10.4}{Notes}{278}

\pagebreak
\chaptit{11}{Bond Percolation in Two Dimensions}{281}
\secttit{11.1}{Introduction}{281}
\secttit{11.2}{Planar Duality}{283}
\secttit{11.3}{The Critical Probability Equals $\frac12$}{287}
\secttit{11.4}{Tail Estimates in the Supercritical Phase}{295}
\secttit{11.5}{Percolation on Subsets of the Square Lattice}{303}
\secttit{11.6}{Central Limit Theorems}{309}
\secttit{11.7}{Open Circuits in Annuli}{314}
\secttit{11.8}{Power Law Inequalities}{324}
\secttit{11.9}{Inhomogeneous Square and Triangular Lattices}{331}
\secttit{11.10}{Notes}{345}

\chaptit{12}{Extensions of Percolation}{349}
\secttit{12.1}{Mixed Percolation on a General Lattice}{349}
\secttit{12.2}{$AB$ Percolation}{351}
\secttit{12.3}{Long-Range Percolation in One Dimension}{351}
\secttit{12.4}{Surfaces in Three Dimensions}{359}
\secttit{12.5}{Entanglement in Percolation}{362}
\secttit{12.6}{Rigidity in Percolation}{364}
\secttit{12.7}{Invasion Percolation}{366}
\secttit{12.8}{Oriented Percolation}{367}
\secttit{12.9}{First-Passage Percolation}{369}
\secttit{12.10}{Continuum Percolation}{371}

\chaptit{13}{Percolative Systems}{378}
\secttit{13.1}{Capacitated Networks}{378}
\secttit{13.2}{Random Electrical Networks}{380}
\secttit{13.3}{Stochastic Pin-Ball}{382}
\secttit{13.4}{Fractal Percolation}{385}
\secttit{13.5}{Contact Model}{390}
\secttit{13.6}{Random-Cluster Model}{393}

\apptit{Appendix I. The Infinite-Volume Limit for Percolation}{397}

\apptit{Appendix II. The Subadditive Inequality}{399}

\apptit{List of Notation}{401}

\apptit{References}{404}

\apptit{Index of Names}{435}

\apptit{Subject Index}{439}



\chapend
