Stochastic Processes on a Lattice and Gibbs Measures
In many domains one encounters "systems" of interacting elements, elements that interact more forcefully the closer they may be. The historical example upon which the theory offered in this book is based is that of magnetization as it is described by the
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MATHEMATICAL PHYSICS STUDIES A SUPPLEMENTARY SERIES TO LETTERS IN MATHEMATICAL PHYSICS
Editors: M. R..ATO, Universite de Dijon, France E. H. LIEB, Princeton University, U.sA. W. THIRRING, Institut fur Theoretische Physik der Universitiit Wien, Austria A. TRAUTMAN, Institute o/Theoretical Physics, Warsaw, Poland
Editorial Board: H. ARAKI, Kyoto University, Japan M. CAHEN, Universite Libre de Bruxelles, Belgium A. CONNES, IH.E.S., France L. FADDEEV, Steklov Institute 0/ Mathematics, Leningrad, U.S.S.R. B. NAGEL, K.T.H., Stockholm, Sweden R. RACZKA, Institut Badan Jadrowych, Warsaw, Poland A. SALAM, International Centre/or Theoretical Physics, Trieste, Italy W. SCHMID, Harvard University, U.SA. J. SIMON, Universite de Dijon, France D. STERNHEIMER, College de France, France I. T.TODOROV, Institute o/Nuclear Research, Sofia, Bulgaria J. WOLF, University o/California, Berkeley, U.S.A.
VOLUME 11
Stochastic Processes on a Lattice and Gibbs Measures BemardPrum and
Jean Claude Fort Universite Paris V, Paris, France
Translated by Bertram Eugene Schwarzbach
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library ofCongress Cataloging-in-Publication Data PrUlI, Bernard. [Processus sur un reseau et mesure de 0lbbs. Engllshl Stochastic Processes an a lattlce and Glbbs IIBasures I by Bernard Prum ana Jean Claude Fort; translated by Bertrall Eugene Schwarzbach. p, CII. -- (Mathematlcal physics studles ; v. 111 Translatlon of: Processus sur un reseau et lIesure de 0lbbs. 1ncludes blbliographlcal references and Index. ISBN 978-94-010-5442-3 ISBN 978-94-011-3268-8 (eBook) DOI 10.1007/978-94-011-3268-8
1. Phase transforllatlons (Statlstlcal physlcsl 2. Is1ng model. 3. Lattlce theory. 4. Markov processes. 5. Mathellatlcal phySics. 1. Fort, Jean Clauae, 1953II. Tltle. III. Ser ies. OCI75.16.P5P7813 1991 530 . 4' 14--d c20
90-23262
ISBN 978-94-010-5442-3
Printed on acid-free paper
AII Rights Reserved © 1991 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1991 So:ftcover reprint ofthe hardcover Ist edition 1991 No part of the material protected by this copyright notice may De reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any infonnation storage and retrieval system, without written pennission from the copyright owner.
Contents Introduction ............................................................................................................ vii Chapter 1 - Aspects of the Ising Model.. ............................................................. 1 1. 2. 3. 4. S.
Introduction ..................................................................................................... 1 Peierls's Argument .........................................................................................4 Imposed Field, Thermodynamic Quantities ................................................. 8 The Unidimensional Case and Tree ........................................................... 12 Antiferromagnetism ...................
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