Random Fields
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534 Chris Preston
Random Fields
Springer-Verlag Berlin· Heidelberg· New York 1976
Author Chris Preston King's College Cambridge/Great Britain
Library of Congress Cataloging in Publication Data
Preston, Christopher J Random fields. (Lecture notes in mathematics; 534) Bibliography: p. Includes index. 1. Stochastic processes.
2.
Measure theory.
3. statistical mechanics. 4. gquilibrium. 1. Title. II. Series: Lecture notes in mathematics (Berlin) ; 534. QA3.L28 vol. 534 [QA274] 5l0'.8s [519.2] 76-26664
AMS Subject Classifications (1970): 28A35, 60GXX, 60K35, 82A05 ISBN 3-540-07852-5 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-07852-5 Springer-Verlag New York' 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 § 54 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 publisher.
© by Springer-Verlag Berlin' Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
In th" last decade there has been a lot of mathematical interest in znodeLs frem c Las s ic aL equi.Lior i urn statistical mechanics; these notes describe some of this work , 'I'hey are concerned, in particular, with the properties of equilibrium states defined in terms of conditional probabilities. This way of defining equi-· states
lS
due to Dobrushin, Lanford ill1d Ruelle; the formulation given
here is due ma.inly to Follmer. The approach taken v i.LL be t'a.i r-Ly abstract, and will be done us i.n., the language and basic techniques of probability theory. It viII thus be assumed that the reader has some
with things like standard measure theory,
con d.itional expectations, the martingale convergence theorem, and probability ke rne.Ls . Some of the deeper results will be obtained using standard Borel apace.i , but no previous knowledge of such obj ects will be required. These notes were written between 1974 and the present; the first
SlX
sections were written in the academic year 1974-75, while the author "as a Fe Ll.ow of Brasenose College, Oxford. The rest was written vh i Le the author wn:; a Fellow of Ki.ng ' s College, Cambridge. The material has been much influence(;' ':ly conversations wi th HeWS Follmer over the last three years, and many t.hanks are due to him.
Chr i s Preston
King's College, Cambr-i.dge February, 1976.
RANDOM FIELDS
1
Section 1.
Introduction
Section 2.
Random fields and specifications
11
Section 3.
Existence of Gibbs states ..
33
Invariant specifications
46
Section 5.
Lattice models
59
section 6.
Continuous models: point processes
87
Section 7.
Specific information gain
.. 111
Section 8.
Some thermodynamics
.. 137
Section 9.
Attractive specifications ..
., 160
Sect
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