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Index

Additive spin systems 174, 256, 282 Annihilating random walks 263 Anti-voter model 162, 167, 176 Attractive spin systems 72, 123, 124,

134-157, 175, 195, 196, 264

Choquet-Deny Theorem 68, 107, 121 Closed linear operator 13 Coalescing random walks 263 Contact process 4, 21, 30, 33, 122,

162, 166, 171, 174,264-314, 315, 325, 328, 353, 419, 460, 462

Core 17 Coupling 64-70, 72, 76, 124-133, 143,

147, 151, 173, 175, 267, 308, 382, 385, 425, 430

Coupling measure 74, 143 Critical exponents 275, 312, 323, 324,

346 Critical values 3, 4, 142, 166, 190,

196, 204, 265-275, 288-289, 307, 312, 344, 356, 357, 459-462

De Finetti's Theorem 365, 369 Dirichlet principle 99 Duality 84-88, 120, 157-176, 230, 245,

266, 363, 430

Ergodic measure 37-38 Ergodic process 3, 12, 31, 129, 136-

138, 163-165,201,265 Ergodic Theorem 37 Exchangeable probability measure

365, 370, 387 Exclusion process 5, 21, 30, 33, 54,

361-417 Exponential convergence 31, 41, 163,

196, 209,211,212,290,302

Feller process 8 Finite range process 39-41 FKG inequality 78, 120

Generator 15, 27, 123, 361, 419 Gibbs states 180-190 Graphical representation 172-175,

176, 276, 283, 294, 383, 404 Griffiths inequalities 186, 222

Harmonic function for a Markov chain 67, 107, 228, 236, 372, 469

Hille-Yosida Theorem 16 Hydrodynamics 415

Invariant measures 10, 17, 52, 94

Linear systems 418-469

Majority vote process 33, 140, 176-177

Markov process 8 Martingale problem 42 Monotone process 72, 134, 383 Monotonicity of measures 71

Nearest-particle systems 312, 315-360 Negative correlations 154, 256, 375,

400, 414

Occupation times for the voter model 262

One-sided spin systems 132, 177 Oriented percolation 294,311

488 Index

Phase transition 3, 185, 196-204, 264 Pointwise ergodic theorem for the

contact process 287 Positive correlations 65, 77-83, 141,

186,231, 240,256,451 Positive definite functions 366, 367,

371, 373 Positive rates conjecture 178, 201 Potential 180 Potlatch process 420, 462, 464, 466 Pregenerator 12, 22

Random walk 68 Reggeon spin model 264, 310 Relative entropy 88-90, 120, 214, 224,

344, 414 Renewal measure 5, 269, 317, 326,

335 Reversible Markov chain 94, 98-106 Reversible measure 91, 94, 192, 318,

335, 365

Semigroup 9 Shape of a growth model

411-412,414-415 264,310,

Smoothing process 420, 461, 464, 466, 467

Spin system 122 Stochastic Ising models 3, 21, 30, 32,

122, 131, 177, 178, 179-225, 309 Subadditive ergodic theorem 277-281,

311,404 Submodular functions 165, 267 Successful coupling 67

Tagged particle 394-402, 417 Transition rates 20

Universality principle 275, 325

Voter model 3, 21, 30, 33, 122, 162, 174, 226-263, 419

Weak convergence of measures 9 Well posed martingale problem 45

Zero range process 466

Postface

When I wrote this book twenty years ago, it was just barely possible to cover the subject of interacting particle systems in one volume. Since then, the subject has grown to the point where such an endeavor would be impossible. As a crude measure of this growth, one might search MathSciNet to determine how many papers contain certain key phrases in the title or review text. Here are some results: "contact process" occurs in 25 papers published prior to 1985, and in 207 papers published since then, while "exclusion process" occurs in 23 papers prior to 1985 and in 308 papers since then. Therefore, this book clearly does not provide anything like a comprehensive treatment of the field as it now exists. On the other hand, it does provide a good place to begin to get into the field and to learn some of the basic techniques that continue to be important today. In addition, it covers the early theory of models that continue to be used and studied in both the mathematics and physics literature. The main ones are the stochastic Ising model, the contact process, and exclusion processes.

The field of interacting particle systems has changed in many respects since the original publication of this book. First, specific problems that were open then have been partially or entirely solved. (There will be more detail about this below.) Secondly, entirely new types of problems have been investigated relating to the models described in this book. (For more on this, see my 1999 book.) Thirdly, new models have been formulated and analyzed. There are many examples, such as multi-type contact processes (where the issue is one of coexistence of multiple types) and reaction diffusion systems (which combine motion with birth and death of particles). Finally, connections have been found and exploited between interacting particle systems and other areas of mathematics. (More on this follows.)

One huge part of the modern theory of interacting particle systems involves the scaling limits of interacting particle systems. This area is known as "hydrodynam­ics", and was in its infancy in 1985. It is treated in the 1999 book of Kipnis and Landim. There is a strong interplay between this set of problems and nonlinear partial differential equations.

Here are a few other examples of connections to various areas of mathematics. Ulam's famous problem on the length L„ of the longest increasing subsequence in a random permutation of {1, ...,n} was found to be closely connected to a variant of the exclusion process in a 1995 paper by Aldous and Diaconis. They

490 Postface

used this connection and the hydrodynamics of this process to give a new proof that Lfi ^ I'sjn. The limiting behavior of interacting particle systems has been related to super Brownian motion (Durrett and Perkins (1999), Cox, Durrett and Perkins (2000)) and to solutions to stochastic partial differential equations (Mueller and Tribe (1995)). One of the important recent advances in probability theory has been the rigorous treatment of random systems whose fluctuations are of order n^l^ rather than the classical n^^^. These systems include random matrices, random permutations, and the one dimensional exclusion process (Johansson (2000), Prahofer and Spohn (2002)).

This book contains statements of 65 problems that were open in 1985. Roughly a third of them have been partially or entirely solved. Perhaps another third have become less important in view of developments during the past two decades. (For example, some were proposed as possible approaches to the solutions of other problems, which have in the meantime been addressed from a different direction.) However, the remainder deserve further attempts at solution. I will conclude this addendum to the book by updating the status of many of these problems. It should be kept in mind that many other interesting and important problems have arisen in the past twenty years, and these are not being mentioned here at all - trying to survey them would be a monumental task.

Chapter I

Problem 4. One approach to solving this problem is to show that the limiting distribution of the process along a sequence of times „̂ f oo is invariant. Versions of this statement have been proved (primarily for one dimensional systems) by Mountford (1995a), Ramirez and Varadhan (1996) and Ramirez (2002). In the case of one dimensional Ising models, convergence to the invariant Gibbs state (with rates) is provided by Holley and Stroock (1989).

Chapter III

Problem 6. The status of the positive rates conjecture for one dimensional spin systems remains unclear. A counterexample to the corresponding statement for a closely related model appears in the paper by Gac (2001). Gac's arguments are very long and complex - the paper runs over 200 pages. To help mitigate this difficulty. Gray (2001) wrote a companion paper, whose purpose is to help the reader understand the main issues and ideas in Gac's paper. The complexity of the proof is such that there has been some dispute about its validity. Gray, who has undoubtedly spent more time on this paper than anyone other than Gacs himself, asserts unequivocally that the proof is correct, while acknowledging the likely existence of mistakes that are "minor and easily fixed". Aside from the question of whether this problem really has been solved, there is also the question of whether the positive rates conjecture is true for attractive spin systems. Gacs' example sheds no light on this issue.

Problem 7. A bit more generally, one can ask for conditions under which an extremal invariant measure for the process must be spatially ergodic. Results in

Postface 491

this direction have been obtained by Andjel (1990) and Andjel and Mountford (1998).

Chapter IV

Problem 4. Ramirez (2002) proves this under reasonable conditions in dimensions 1 and 2.

Chapter VI

Problem 1. The upper bound "kc < 1.942 was proved by Liggett (1995).

Problem 2. The extinction of the critical contact process was proved in all dimen­sions by Bezuidenhout and Grimmett (1990).

Problem 6. The central limit theorem for the right edge was proved by Galves and Presutti (1987). A simpler proof was later given by Kuczek (1989) in the context of oriented percolation.

Problem 9. These extensions were proved by Bezuidenhout and Grimmett (1990).

Problem 10. This follows from the construction in Bezuidenhout and Grimmett (1990).

Chapter VII

Problem 2. Some information on these asymptotics was provided by Liggett (1987).

Problems 7 and 17. Mountford (1997) proves the complete convergence theorem for supercritical attractive reversible nearest particle systems (under a mild as­sumption on y3.) His 2002 paper proves a substantially weaker result in the more difficult critical case.

Problems 13,14 and 16. In his two 1992 papers, Mountford proves that the critical value is 1 for the centered and uniform nearest particle systems (the ones with rates given by (5.16) and (5.17)).

Problem 18. Mountford (1995b) proves exponential convergence of the process to So under some assumptions.

Chapter VIII

Problem 4. Converence to a product measure is proved in Mountford (2001) for one dimensional systems with nonzero drift. The assumption on the initial distribution is somewhat weaker than ergodicity.

Problem 6. This characterization was proved under extra hypotheses on the transition probabilities by Bramson, Liggett and Mountford (2002).

Problem 7. The statement of this problem is rather vague, so while significant progress has been made on it, there is a lot of room for further work here. Many of the hydrodynamical results that have been proved for the exclusion process yield

492 Postface

partial solutions to this problem - see the 1999 books by Kipnis and Landim, and Liggett and the references there. For the nearest neighbor process in one dimension, the specific problem mentioned here in the case A + p = 1,X < 1/2, was handled in Andjel, Bramson and Liggett (1988).

Problem 8. This asymptotic normality was was proved by Varadhan (1995) for asymmetric systems with mean zero and by Sethuraman, Varadhan and Yao (2000) for systems in three and higher dimensions with drift.

Problem 9. This statement was proved by Saada (1987).

Chapter IX

Problem 1. Apparently this problem is still open in the context of linear systems. However, it has been settled (in the negative) for systems of branching Brownian motions by Bramson, Cox and Greven (1993) (for binary branching) and by Klenke (1998) (for branching laws with large tails).

Problem 4. This extremality follows from results in Andjel (1990).

Problem 5. Convergence to equilibrium is proved, even for some initial distribu­tions that are not translation invariant, in Cox, Klenke and Perkins (2000).

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Corrections for "Interacting Particle Systems" by Thomas M. Liggett

Page 33, line 9 from the bottom. Change braces to parentheses: crir], dt;).

Page 82, line 6 from the bottom. Change the n on the main line to an 77:

G{r}) = E'^g(r],r}s^,...,r]s,^_^).

Page 111, display (8.11). The symbol under the summation should be x instead of y:

(8.11) fi(y) = J2^(x)a(x~y).

Page 131, hne 18. There is a missing parenthesis:

sup \c(x, §) - c(x, ^u)\ > sup \c(x, Y]) - c{x, r]u)\

Page 171, second line following the display in Example 5.33: that should be than:

dimensional contact process. For k larger than its critical value, the one-

Page 185, line 3. Change the last ^ to y:

(1.17) yr2,K- \r] = yon T2\Ti) = vri,y(-)

Page 204, line 5. The display should read:

2v{r] : r]{x) =. 1} - 1 = [1 - (sinh2/3)-4]^/^

Page 220, line 3 from the bottom. The subscript on a should be T^:

lim -^y]aT^(x) = 0. xeTn

496 Errata

Page 229, line 7. Change Chapter I to Chapter 11.

Page 231, display (1.14). The final A should have bars around it:

(1.14) giA) = P^(\At\ < \A\ for some t > 0).

Page 232. line 6. ... greater than or equal...

Page 241, line 11. The final symbol should be 0 rather than 0:

(c) At = BfU Ct for t < T, where r is the first time that Bt HCt ^ 0.

Page 275, line 10. ... At this point, we know that...

Page 280, bottom line. Add a comma:

X^„ = max(X^,„, -N(n - m)).

Page 312, line 7 from the bottom. ... Of course a major difficulty in proving the ...

Page 370, (b) of Lemma 1.18. There is a missing":

(b) lim^^oo U2(t)g2(x) = 0 for all x e S^, and

Page 395, fines 4, 7 and 8. Change A e S\{0} to A c S\{0}.

Page 411. The top display should read:

0 < f^ilv : r](x) = 1, r](y) = 1} - [Mi{r] : r](x) = 1, r](y) = 1}

< [fi2{r] : rj(x) = 1} - /xi{/7 : r](x) = 1}]

+ [M2{^ : r](y) = 1} - fii{r] : r](y) = 1}].