First-Passage Percolation on the Square Lattice
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671 R. T. Smythe John C. Wierman
First-Passage Percolation on the Square Lattice
Springer-Verlag Berlin Heidelberg New York 1978
Authors R. T. Smythe Department of Mathematics University of Oregon Eugene, OR 91403/USA
John C. Wierman School of Mathematics 127 Vincent Hall University of Minnesota Minnesota, MN 55455/USA
Library or Congress Cataloging in Publication Data
Main entry under title:
First-passage percolation on the square lattice. (Lecture notes in mathematics ; 671) Bibliography: p. Includes index. 1. Limit theorems (Probability theory) 2. Renewal theory. 3. Matrices. I. Smythe, Robert T., 1941II. Wierman, John C., 1949III. Title: Percolation on the square lattice. IV. Series: Lecture notes in mathematics (Berlin) ; 671. QA3.L28 no. 671 [QA273.67J 510'.8s [519.2J 78-13679 AMS Subject Classifications (1970): 60F15, 60K05, 60K99, 94A20
ISBN 3-540-08928-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08928-4 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978
2141/3140-543210
Preface
The mathematical study of percolation processes has now been going on for some twenty years.
Most of this work has concentrated
on the case in which the bonds (or sites) have only two states, open or closed; we have called this the Bernoulli percolation model.
First-
passage percolation theory, initiated by Hammersley and Welsh in 1965, may be viewed as a generalization of Bernoulli percolation, permitting the passage times along bonds to have any non-negative distribution with a finite mean.
There are in the literature a number of expositions
of various aspects of Bernoulli percolation, including those of Shante and Kirkpatrick (1971), Essam (1972), Welsh (1977), and Seymour and Welsh (1977).
By contrast, much of the work on first-passage percolation
is quite recent and no reasonably comprehensive account has been published. The present work is an attempt to fill this gap.
Because our
principal interest is in first-passage percolation, no attempt has been made to provide an exhaustive treatment of Bernoulli percolation (we have, however, presented most of the known rigorous results in Chapter III).
Again, since our emphasis is primarily mathematical,
we have not attempted to summarize the large literature on Monte Carlo studies of percolation, but have contented ourselves with a few references in section 3.10.
Finally, we have restricted consi.deration to
the square lattice, largely for reasons of mathematical tractability. It will be clear, however, that a number of the results or techn
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