Mean Field Models for Spin Glasses Volume I: Basic Examples
This is a new, completely revised, updated and enlarged edition of the author's Ergebnisse vol. 46: "Spin Glasses: A Challenge for Mathematicians". This new edition will appear in two volumes, the present first volume presents the basic results and method
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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 54
Michel Talagrand
Mean Field Models for Spin Glasses Volume I: Basic Examples
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 first part of a treatise on Spin Glasses in the series Ergebnisse der Mathematik und ihrer Grenzgebiete. The second part is Vol. 55 of the Ergebnisse series. The first edition of the treatise appeared as Vol. 46 of the same series (978-3-540-00356-4). ISBN 978-3-642-15201-6 e-ISBN 978-3-642-15202-3 DOI 10.1007/978-3-642-15202-3 Springer Heidelberg Dordrecht London New York Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISSN 0071-1136 Mathematics Subject Classification (2010): Primary: 82D30, 82B44. Secondary: 82C32, 60G15, 60K40 © 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: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Wansoo T. Rhee, for so many reasons.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI 1.
The 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14
Sherrington-Kirkpatrick Model . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Notations and Simple Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Gaussian Interpolation and the Smart Path Method . . . . . . . . 12 Latala’s Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A Kind of Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . 51 The Cavity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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