Stochastic Interacting Systems: Contact, Voter and Exclusion Processes

345 Pages·1999·7.195 MB·English

by  Thomas M. Liggett (auth.)

Gruncllehren cler
mathematischen Wissenschaften 324
A Series of Comprehensive Studies in Mathematics
Editors
S. S. Chern B. Eckmann P. de la Harpe
H. Hironaka F. Hirzebruch N. Hitchin
L. Hormander M.-A. Knus A. Kupiainen
J. Lannes G. Lebeau M. Ratner D. Serre
Ya.G. Sinai N. J. A. Sloane J.Tits
M. Waldschmidt S. Watanabe
Managing Editors
M. Berger J. Coates S.R.S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Thomas M. Liggett
Stochastic
Interacting Systellls:
Contact, Voter and
Exclusion Processes
With 6 Figures
Springer
Thomas M. Liggett
Mathematics Department
University of California
Los Angeles, CA 90095-1555
USA
email: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Liggett, Thomas M.: Stochastic interacting systems: contact, voter and
exclusion processes / Thomas M. Liggett. - Berlin; Heidelberg; New York;
Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo:
Springer 1999
(Grundlehren der mathematischen Wissenschaften; 324)
Mathematics Subject Classification (1991): 60K35
ISSN 0072-7830
ISBN 978-3-642-08529-1 ISBN 978-3-662-03990-8 (eBook)
001 10.1007/978-3-662-03990-8
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Preface
Interacting particle systems is a branch of probability theory that has rich con
nections with a number of areas of science - primarily physics in the early days,
but increasingly biology and the social sciences today. Stochastic processes of the
sort that are studied in this field are used to model magnetism, spatial competition,
tumor growth, spread of infection, and certain economic systems, to mention but
a few of the many areas of application.
The subject is by now about thirty years old. At the midpoint of that thirty year
period, I wrote the book Interacting Particle Systems (IPS) as an attempt to give
some order to the work that had been done by then, and to make the field more
accessible to new researchers, and more useful to workers in areas of application.
Judging from the rapid development of the field since then, this attempt appears
to have been successful.
My earlier book covered more or less the entire field, as it was at that time.
Even so, some topics, such as zero range processes and the then emerging area
of hydrodynamics, were mentioned only briefly. By now, the field has grown
to the point where it would be impossible to cover it entirely in one book. In
fact, a number of books that treat special topics within the field have appeared in
the interim - see for example Chen (1992), DeMasi and Presutti (1991), Durrett
(1988), Kipnis and Landim (1999), Konno (1994), and Spohn (1991).
IPS was organized horizontally, in that a separate chapter was devoted to each
type of model: stochastic Ising models, voter models, contact processes, nearest
particle systems, exclusion processes, and linear systems. The present book has a
more vertical appearance. It takes but three of these models - the ones given in
the title - and traces their development since 1985. Nearest particle systems are
omitted because, even though substantial progress has been made on them since
1985 (especially by T. Mountford), they are by their nature somewhat special.
Linear systems are omitted because they have been less active recently, while
stochastic Ising models are omitted because developments in that area alone would
justify an entire book.
Even my relatively modest objective of covering recent work on three models
cannot be attained in a book of reasonable size, so I have had to make some
choices about what to include. These choices reflect to some extent, of course,
my own interests and perspective on the field. Other work on these models is
described briefly in the Notes and References section for each of the three parts.
VI Preface
I have tried to make the treatment as self-contained as possible without dupli
cating too much of the contents of IPS. The initial section on background material,
as well as the initial Preliminaries section of each of the three main parts, should
help in this regard.
This book is an outgrowth of my Wald Memorial Lectures - see Liggett (1997)
- that also dealt primarily with contact, voter and exclusion processes. One of the
advantages of this selection of topics is that it provides illustrations of the use of
some of the most important tools in the area: percolation and graphical techniques
(Part I), correlation inequalities (Part I), duality (Parts I and II), coupling (Parts I
and III), and partial differential equations (Part III). It should not be expected that
many models that come up in applications will fit exactly into one of the three
classes we consider here. The hope, rather, is that a good understanding of the
behavior of these classes and of the tools used in their analysis will facilitate the
analysis of new models that arise.
It should be clear from the above comments that the present book is in no
sense a second edition of IPS. It is also not really a second volume. I hope it will
playa role similar to that of IPS, though, as a reference for workers in probability
and areas of application, as well as an advanced text. There is plenty of material
in it for a semester course, and by adding lectures based on the papers discussed
in the Notes and References sections, it can easily be extended to a full year. The
mathematical prerequisites for reading this book are year-long courses in analysis
and probability - a probability course based on Durrett (1996), for example.
We tum now to a brief survey of the contents of this book. The contact process
has been one of the central models in the subject since its introduction by T. Harris
over twenty years ago. The theory as of 1985 was primarily one dimensional. Very
little was known in higher dimensions. Developments during the past decade have
therefore rendered much of Chapter VI of IPS largely out of date. The primary
exception to this is the first section, on critical value bounds. Here only minor
improvements have been made. As a consequence, our treatment of the contact
process in the first part of this book starts almost from the beginning. The critical
value bounds from IPS are stated in the Preliminaries section. While the proofs
from IPS are not repeated, a more elaborate version of the argument plays a
dominant role in Part II of this book.
Until the early 1990's, the contact process was studied almost exclusively on
the d-dimensional integer lattice Zd. Sections 2 and 3 of Part I explain some of
these developments. Section 2 is primarily devoted to the advances by Bezuiden
hout and Grimmett (1990, 1991) that more or less completed the Zd theory,
showing among other things that the critical contact process dies out. Section 3
is dedicated to several results that address the following natural question: Since
real systems are finite, and contact processes on finite sets die out with probability
one, how can the phase transitions that occur on infinite sets have any bearing on
our understanding of real systems?
Section 4 traces the development of the theory of contact processes on ho
mogeneous trees. Interest in this comes from the fact that an intermediate phase
Preface VII
occurs in this context that is absent in the case of Zd. Briefly, the contact process
on Zd has one critical value, while the contact process on a homogeneous tree
(other than Z 1) has two distinct critical values. Between these two critical values,
the finite process survives globally, but dies out locally. Unlike Zd, the tree is
large enough that the infected set can wander out to infinity without dying out,
but this can only happen for intermediate values of the infection parameter.
The story is quite different for voter models. The voter models discussed in
Chapter V of IPS are what are now known as linear voter models. Their ergodic
theory was more or less completed in IPS. While significant progress has been
made on linear voter models since then, and is discussed in the Notes and Refer
ences section, the focus of Part II is on their nonlinear cousins. Nonlinear voter
models require quite a different approach, primarily because their duals (when they
exist) are harder to analyze. While the theory of nonlinear voter models is still
very far from being complete, there are close connections to the contact process,
and this makes it a natural candidate for inclusion in this book. The main theorem
in Part II gives a complete classification of threshold voter models with threshold
level = 1. The proof given there is a substantial improvement over my original
treatment, which was computer aided and contained a serious error.
The situation for exclusion processes is again different. The material in Chapter
VIII of IPS has in general not been superceded by subsequent developments.
However, there is a whole new collection of issues that have been investigated,
and it is to these that we address our attention in the final part of this book.
We again omit a treatment of the by now mature area of hydrodynamics, partly
because it is well covered in the books of De Masi and Presutti (1991), Spohn
(1991), and Kipnis and Landim (1999), and partly because it has quite a different
flavor from the topics we will cover here.
The first main section of Part III gives a probabilistic treatment of shocks in the
asymmetric, nearest neighbor, exclusion process in one dimension, based on work
by Ferrari and his coauthors. The main technique used here is coupling. Then we
move to a more analytic treatment of roughly the same issues that was developed
by Derrida and his coworkers. This is known as the matrix approach. Finally,
we turn to central limit theorems for tagged particles in more general exclusion
processes, based on work of Varadhan and coauthors. IPS has a treatment of
this only in the case of the symmetric, nearest neighbor, one-dimensional system,
which has a different behavior than the general system considered here.
The Background and Tools section at the beginning of the book describes the
basic particle system setup, and some of the key techniques that are useful in the
analysis of many models - coupling, monotonicity, correlation inequalities and
subadditivity, for example. These first few subsections should be read before ven
turing into the book proper, but the latter subsections can be skipped, and read
when they are used later on. Each of the three parts begins with a brief description
of that particular model, and gives precise statements of results from the corre
sponding chapters ofIPS (or from other references) that are used later. With this
exception, the numbered sections within each part are largely self-contained. Each
VIII Preface
part ends with a Notes and References section that has two functions. First, it
details the sources of the material in that part. Secondly, it contains brief descrip
tions of the large amount of related work that I have not been able to include in
the book itself.
While this book is more or less self-contained, the reader may find that reading
parts of IPS first makes the going easier. Here are my suggestions about what parts
of IPS to read in this case:
(a) The first four sections of Chapter I and the first three sections of Chapter II
before starting this book.
(b) The first three sections of Chapter VI before reading Part I.
(c) The first two sections of Chapter V before reading Part II.
(d) The first three sections of Chapter VIII before reading Part III.
A popular (I think) feature of IPS was its sets of open problems. I have not
attempted to do anything formal of this sort here. There are simply too many
open problems, and many of them are not directly about the three types of models
I treat here, but rather about other models that are nevertheless closely related
to contact, voter and exclusion processes. However, I do mention problems that
I think should be looked at when they arise naturally, mainly in the Notes and
References sections.
As I mentioned in the preface to IPS, my wife Chris had a lot to do with my
writing that book. For the last several years, she has been lobbying for a follow-up.
It took a while, but she finally got it. In the earlier preface, I mentioned some of
the people who had had the most impact on my work, as well as on the subject as
a whole. Most have continued to be leaders in the field, but they have now been
joined by a large and impressive group of younger mathematicians. I won't list
them here, but most appear prominently in the bibliography. One of the measures
of a field of research is the caliber of researcher that it attracts. By this measure,
interacting particle systems has been a great success.
Pablo Ferrari, Norio Konno, Tom Mountford, Roberto Schonmann, and espe
cially my former students, Amber Puha and Li-Chau Wu, have read parts of this
book, and made suggestions for improvement - I very much appreciate their input.
I would like to acknowledge the National Science Foundation for its support of
my work over the past quarter of a century, and the Guggenheim Foundation for
freeing my time in 1997-98, so that I could devote much of it to writing this book.
Without their support, this work would not have been possible.
Los Angeles, CA Thomas M Liggett
March I, 1999
Contents
Background and Tools
The Processes
Invariant Measures 4
Reversible Measures 5
Coupling, Monotonicity and Attractiveness 6
Correlation Inequalities 8
Duality ...... . 11
Subadditivity 12
Oriented Percolation 13
Domination by Product Measures 14
Renewal Sequences and Logconvexity 16
Translation Invariant Measures 21
Some Ergodic Theory 22
Branching Processes 25
Some Queuing Theory 26
The Martingale CLT 29
Part I. Contact Processes 31
1. Preliminaries 31
Description of the Process 31
The Graphical Representation; Additivity 32
The Upper Invariant Measure 34
Duality ........... . 35
Convergence ........ . 36
Monotonicity and Continuity in A 38
Rate of Growth ......... . 40
Survival and Extinction; Critical Values 42
Preview of Part I ........... . 44
2. The Process on the Integer Lattice Zd 44
The Boundary of a Big Box Has Many Infected Sites 45
The Finite Space-Time Condition ........ . 50
Comparison with Oriented Percolation ..... . 51
First Consequences of the Percolation Comparison 54
X Contents
Exponential Bounds in the Supercritical Case 57
Exponential Decay Rates in the Sub critical Case 60
A Critical Exponent Inequality 69
3. The Process on {I, ... , N}d 71
The Subcritical Case . 72
The Supercritical Case ... 74
4. The Process on the Homogeneous Tree Td 78
Some Critical Value Bounds ....... 79
Branching Random Walk . . . . . . . 80
Back to the Contact Process - the Function ¢ 86
Extinction at the First Critical Value ..... 91
Existence of an Intermediate Phase 94
The Sequence u and its Growth Parameter f3()...) 96
The Complete Convergence Theorem 103
Continuity of the Survival Probability 104
The Growth Profile .......... 105
Invariant Measures in the Intermediate Regime - First Construction 109
Invariant Measures in the Intermediate Regime - Second Construction 119
Strict Monotonicity of f3()...) 123
5. Notes and References 125
Part II. Voter Models 139
1. Preliminaries 139
Description of the Process 139
Clustering and Coexistence 140
The Linear Voter Model .. 140
The Threshold Voter Model 142
The Graphical Representation 142
Duality when T = 1 143
Preview of Part II ...... 145
2. Models with General Threshold and Range 146
Fixation for Large Thresholds ....... 146
Clustering in One Dimension ........ 147
Coexistence; the Threshold Contact Process 151
The Threshold Contact Process with Large Range 153
The Threshold Voter Model with Large Range 155
3. Models with Threshold = 1 ........... 155
Duality for the Threshold Contact Process, T = 1 156
Reduction to One Dimension 158
The Convolution Equation 159
The Density .......... 162