Percolation Theory for Mathematicians
Quite apart from the fact that percolation theory had its orlgln in an honest applied problem (see Hammersley and Welsh (1980)), it is a source of fascinating problems of the best kind a mathematician can wish for: problems which are easy to state with a
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Harry Kesten
Percolation Theory for Mathematicians
Progress in Probability and Statistics Vol. 2 Edited by P. Huber and M. Rosenblatt
Springer Science+Business Media, LLC
Harry Kesten
Percolation Theory for Mathematicians
1982
Springer Science+Business Media, LLC
Author: Harry Kesten Department of Mathematics Cornell University Ithaca, NY 14853
Library of Congress Cataloging in Publication Data Kesten, Harry, 1931Percolation theory for mathematicians. (Progress in probability and statistics; v. 2) Bibl iography: p. Includes index. 1. Percolation (Statistical physics) I. Title. II. Series. QC174.85.P45K47 1982 530.1 '3 82-19746 ISBN 978-0-8176-3107-9
CIP-Kurztitelaufnahme der Deutschen Bibliothek Kes ten, Harry: Percolation theory for mathematicians / Harry Kesten. (Progress in probability and statistlcs ; Vol. 2) ISBN 978-0-8176-3107-9 ISBN 978-1-4899-2730-9 (eBook) DOI 10.1007/978-1-4899-2730-9 NE: GT
All rights reserved. No part of this publication may be ~eproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, methanical, photocopying, recording or otherwise, without prior permission of the copyright owner. © Springer Science+Business Media New York 1982 Originally published by Birkhauser Boston, Inc. in 1982 ISBN 978-0-8176-3107-9
PREFACE Quite apart from the fact that percolation theory had its orlgln in an honest applied problem (see Hammersley and Welsh (1980)), it is a source of fascinating problems of the best kind a mathematician can wish for: problems which are easy to state with a minimum of preparation, but whose solutions are (apparently) difficult and require new methods. At the same time many of the problems are of interest to or proposed by statistical physicists and not dreamt up merely to demons~te ingenuity. Progress in the field has been slow. Relatively few results have been established rigorously, despite the rapidly growing literature with variations and extensions of the basic model, conjectures, plausibility arguments and results of simulations. It is my aim to treat here some basic results with rigorous proofs. This is in the first place a research monograph, but there are few prerequisites; one term of any standard graduate course in probability should be more than enough. Much of the material is quite recent or new, and many of the proofs are still clumsy. Especially the attempt to give proofs valid for as many graphs as possible led to more complications than expected. I hope that the Applications and Examples provide justification for going to this level of generality. I taught a graduate course on this material at Cornell University in Spring 1981, but the beginning of the monograph was a set of notes for a series of lectures at Kyoto University, Japan, which I visited in summer 1981 on a Fellowship of the Jaran Society for the Promotion of Science. I am indebted to a large number of people for helpful discussions. I especially value various suggestions made by J.T. Cox, R. Durrett, G.R. Grimmett and S. Kotani. I also wish to
