Mean Field Models for Spin Glasses Volume II: Advanced Replica-Symme
This is a new, completely revised, updated and enlarged edition of the author's Ergebnisse vol. 46: "Spin Glasses: A Challenge for Mathematicians" in two volumes (this is the 2nd volume). In the eighties, a group of theoretical physicists introduced sever
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A Series of Modern Surveys in Mathematics
Editorial Board G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollár, Princeton G. Laumon, Orsay H. W. Lenstra, Jr., Leiden S. Müller, Bonn J. Tits, Paris D. B. Zagier, Bonn G. Ziegler, Berlin Managing Editor R. Remmert, Münster
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Volume 55
Michel Talagrand
Mean Field Models for Spin Glasses Volume II: Advanced Replica-Symmetry and Low Temperature
Michel Talagrand Université Paris 6 Institut de mathématiques UMR 7586 CNRS Place Jussieu 4 75252 Paris Cedex 05 France [email protected]
This volume is the second part of a treatise on Spin Glasses in the series Ergebnisse der Mathematik und ihrer Grenzgebiete. The first part is Vol. 54 of the Ergebnisse series (ISBN 978-3-642-15201-6). The first edition of the treatise appeared as Vol. 46 of the same series (978-3-540-00356-4). ISSN 0071-1136 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISBN 978-3-642-22252-8 e-ISBN 978-3-642-22253-5 DOI 10.1007/978-3-642-22253-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011936027 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Giorgio Parisi, for the new territories he discovered.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI Part I. Advanced Replica-Symmetry 8.
The 8.1 8.2 8.3 8.4 8.5
Gardner Formula for the Sphere . . . . . . . . . . . . . . . . . . . . . 3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Gardner Formula for the Gaussian Measure . . . . . . . . . . . 13 The Gardner Formula for the Sphere . . . . . . . . . . . . . . . . . . . . . 24 The Bernoulli Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
9.
The 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9
Gardner Formula for the Discrete Cube . . . .
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