Probability on Discrete Structures

362 Pages·2004·4.13 MB·English

by  David Aldous

Encyclopaedia of Mathematical Sciences
Volume no
Probability Theory
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Harry Kesten (Editor)
Probability
on Discrete
Structures
Springer
Harry Kesten
Cornell University Department of Mathematics
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Mathematics Subject Classification (2000): 6oB99, 6oCo5, 6oF17, 6oG5o,
6oJio, 60J27, 6oK35
ISSN 0938-0396
ISBN 3-540-00845-4 Springer-Verlag Berlin Heidelberg New York
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Preface
Probability on discrete structures covers a wide area. Most probability prob-
lems involve random variables indexed by space and/or time. Almost always
these problems have a version in which space and/or time are taken discrete.
Roughly speaking this volume deals with some areas in which the discrete
version is more natural than the continuous one, or perhaps even the only one
which can be formulated without complicated constructions and machinery.
Clear examples of this situation can be found in the articles in this volume on
the random cluster model (by Grimmett) and on first-passage percolation (by
Howard) and in most of the problems in the forthcoming book "Probability
on Trees and Networks" by R. Lyons and Y. Peres. The article by Howard
actually also discusses a continuous variant - called Euclidean first-passage
percolation - but this came later and even though this continuous version has
some clear advantages, its analysis brings in extra difficulties.
Problems on discrete structures can often be stated with minimal prereq-
uisites, and sometimes only "elementary" (but by no means easy) probability
theory is needed for their solution. Often the arguments have more of a com-
binatorial flavor than an analytic one, but the articles here certainly do not
shun the use of the tools of analysis.
Since the subject matter of this volume is so broad and varied, it is not
surprising that it does not lend itself to a simple linear ordering. It did not
seem possible to me to produce a volume which introduced a reader to much
of the field in textbook fashion, in which one goes through the chapters in
order. Instead, the present volume introduces a reader to the problems and
progress so far, in various representative directions and subjects in which there
is considerable activity, and which have seen recent successes. The various
articles are not dependent on each other.
There is one obvious omission from the list of possible topics in this vol-
ume, namely percolation. This subject was omitted here, because its classical
aspects have been reviewed only two years ago by G. Grimmett in an ency-
clopedia article (see Development of Mathematics, 1950-2000, Jean-Paul Pier
VI Preface
ed.), while its very recent successes by Lawler, Schramm, Smirnov and Werner
are still evolving while this volume is being prepared.
I hope this volume will give a reader a solid introduction to the flavor and
excitement of probability on discrete structures and encourage her to work in
the subject herself.
Ithaca, NY Harry Kesten
July 25, 2003
Contents
The Objective Method: Probabilistic Combinatorial
Optimization and Local Weak Convergence
David Aldous, J. Michael Steele ....................................
1
The Random-Cluster Model
Geoffrey Grimmett ............................................... 73
Models of First-Passage Percolation
C. Douglas Howard............................................... 125
Relaxation Times of Markov Chains
in Statistical Mechanics and Combinatorial Structures
Fabio Martinelli ................................................. 175
Random Walks on Finite Groups
Laurent Saloff-Coste.............................................. 263
Index .......................................................... 347

List of Contributors
David Aldous Fabio Martinelli
Department of Statistics, Dipartimento di Matematica,
University of California Berkeley, Universita' di Roma 'Ire,
Berkeley, CA, 94720-3860. L.go S. Murialdo 1,
[e-mail] 00146 Roma, Italy.
[e-mail]
Geoffrey Grimmett
Statistical Laboratory, Laurent Saloff-Coste
University of Cambridge, Cornell University,
Wilberforce Road, Department of Mathematics,
Cambridge CB3 OWB, Ithaca, NY 14853-4201.
United Kingdom. [e-mail]
g.r.grimmett@
statslab.cam.ac.uk J. Michael Steele
Department of Statistics,
C. Douglas Howard Wharton School,
City University of New York, University of Pennsylvania,
Baruch College. Philadelphia, PA 19104-6302.
[e-mail] [e-mail]