Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory

Simple random walks - or equivalently, sums of independent random vari­ ables - have long been a standard topic of probability theory and mathemat­ ical physics. In the 1950s, non-Markovian random-walk models, such as the self-avoiding walk,were introduce

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w. BeiglbOck R.P.Geroch E.H.Lieb T.Regge W. Thirring Series Editors

Roberto Fernandez

JUrg Frohl ich AI an D. Sakal

Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory With 26 Figures

Springer-Verlag Berlin Heidelberg GmbH

Dr. Roberto Fernandez Professor Jiirg Frohlich Institut flir Theoretische Physik, ETH Hiinggerberg CH-8093 Zurich, Switzerland

Professor Alan D. Sokal Department of Physics, New York University 4, Washington Place, New York, NY 10003, USA

Editors Wolf Beiglbock

Tullio Regge

Institut flir Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 294 W-6900 Heidelberg 1, FRO

Instituto di Fisica Teorica Universita di Torino, C. so M. d' Azeglio, 46 1-10125 Torino, Italy

Robert P. Geroch Enrico Fermi Institute University of Chicago 5640 Ellis Ave. Chicago, IL 60637, USA

Walter Thirring Institut flir Theoretische Physik der Universitat Wien Boltzmanngasse 5 A-I090 Wien, Austria

Elliott H. Lieb Jadwin Hall P. O. Box 708 Princeton University Princeton, NJ 08544-0708, USA

ISBN 978-3-662-02868-1 Library of Congress Cataloging-in-Publication Data. Fernandez, R. (Roberto), 1951-. Random Walks, critical phenomena, and triviality in quantum field theory / R. Fernandez, J. Frohlich, A. D. Sokal. p. cm. -- (Texts and monographs in physics). Includes bibliographical references and index. 1. Quantum

ISBN 978-3-662-02868-1 ISBN 978-3-662-02866-7 (eBook) DOI 10.1007/978-3-662-02866-7 field theory. 2. Random walks (Mathematics). 3. Critical phenomena (Physics). I. Frohlich, J. (JUrg.), 1946-. II. Sokal, A. D. (Alan D.), 1955-. III. Title. IV. Series. QCI74.45.F46 1991 530.1'43--dc20 91-40850 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 ways, 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-Verlag Berlin Heidelberg GmbH, Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Originally published by Springer-Verlag Berlin Heidelberg New York in 1992 Softcover reprint of the hardcover I Sl edition 1992 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.

Typesetting: Camera ready by authors Production Editor: P. Treiber 55/3140-543210 - Printed on acid-free paper

Foreword

Simple random walks - or equivalently, sums of independent random variables - have long been a standard topic of probability theory and mathematical physics. In the 1950s, non-Markovian random-walk models, such as the self-avoidin