Additive and Cancellative Interacting Particle Systems
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David Griffeath
Additive and Cancellative Interacting Particle Systems
Springer-Verlag Berlin Heidelberg New York 1979
Author David Griffeath Dept. of Mathematics University of Wisconsin Madison, Wl 53706 USA
AMS Subject Classifications (1970): 60 K35 ISBN 3-540-09508-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09508-X Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under £354 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publishel © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface These notes are based on a course given at the University of Wisconsin in the spring of 1978.
The subject is (stochastic) interacting particle systems, or
more precisely, certain continuous time M a r k o v processes with state space S = {all subsets of Z d } .
This area of probability theory has been quite active
over the past ten years : a list of references, by no means comprehensive, found at the end of the exposition.
m a y be
In particular, several surveys on related
material are already available, a m o n g them Spitzer (1971), D a w s o n
(1974b),
Spitzer (1974b), Sullivan (1975), Georgii (1976), Liggett (1977) and Stroock (1978). There is rather little overlap between the present treatment and the above articles, and where overlap occurs our approach is s o m e w h a t different in spirit. Specifically, these notes are based on 9raphical representations of particle systems, an approach due to Harris (1978).
The basic idea is to give explicit
constructions of the processes under consideration with the aid of percolation substructures.
While limited in applicability to those systems which admit such
representations, Harris' technique manages to handle a large number of interesting models.
W h e n it does apply, the graphical approach has several advantages over
alternative methods.
First, since the systems are constructed from "exponential
alarm clocks, " the existence problem does not arise.
Also, the uniqueness problem
can be handled with m u c h less difficulty than for more general particle systems. Another appealing feature is the geometric nature of the representation, which leads to "visual" probabilistic proofs of m a n y results. coupling.
Finally, there is the matter of
O n e of the basic strategies in studying particle systems is to put two or
more processes on a joint probability space for comparison purposes.
Graphical
representations have the property that processes starting from arbitrary initial configurations are all defined on the same probability space, in such a w a y tha
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