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This IMA Volume in ~athematics and its Applications PERCOLATION
THEORY AND ERGODIC THEORY OF INFINITE PARTICLE SYSTEMS represents
the proceedings of a workshop which was an integral part of the
19R4-85 IMA program on STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR
APPLICATIONS We are grateful to the Scientific Committee: naniel
Stroock (Chairman) Wendell Fleming Theodore Harris Pierre-Louis
Lions Steven Orey George Papanicolaoo for planning and implementing
an exciting and stimulating year-long program. We especially thank
the Workshop Organizing Committee, Harry Kesten (Chairman), Richard
Holley, and Thomas Liggett for organizing a workshop which brought
together scientists and mathematicians in a variety of areas for a
fruitful exchange of ideas. George R. Sell Hans Weinherger PREFACE
Percolation theory and interacting particle systems both have seen
an explosive growth in the last decade. These suhfields of
probability theory are closely related to statistical mechanics and
many of the publications on these suhjects (especially on the
former) appear in physics journals, wit~ a great variahility in the
level of rigour. There is a certain similarity and overlap hetween
the methods used in these two areas and, not surprisingly, they
tend to attract the same probabilists. It seemed a good idea to
organize a workshop on "Percolation Theory and Ergodic Theory of
Infinite Particle Systems" in the framework of the special
probahility year at the Institute for Mathematics and its
Applications in 1985-86. Such a workshop, dealing largely with
rigorous results, was indeed held in February 1986.
Grimmett, Geoffrey: Percolation and disordered systems.- Kesten,
Harry: Aspects of first passage percolation. "
Most probability problems involve random variables indexed by space
and/or time. These problems almost always have a version in which
space and/or time are taken to be discrete. This volume deals with
areas in which the discrete version is more natural than the
continuous one, perhaps even the only one than can be formulated
without complicated constructions and machinery. The 5 papers of
this volume discuss problems in which there has been significant
progress in the last few years; they are motivated by, or have been
developed in parallel with, statistical physics. They include
questions about asymptotic shape for stochastic growth models and
for random clusters; existence, location and properties of phase
transitions; speed of convergence to equilibrium in Markov chains,
and in particular for Markov chains based on models with a phase
transition; cut-off phenomena for random walks. The articles can be
read independently of each other. Their unifying theme is that of
models built on discrete spaces or graphs. Such models are often
easy to formulate. Correspondingly, the book requires comparatively
little previous knowledge of the machinery of probability.
Most probability problems involve random variables indexed by space and/or time. These problems almost always have a version in which space and/or time are taken to be discrete. This volume deals with areas in which the discrete version is more natural than the continuous one, perhaps even the only one than can be formulated without complicated constructions and machinery. The 5 papers of this volume discuss problems in which there has been significant progress in the last few years; they are motivated by, or have been developed in parallel with, statistical physics. They include questions about asymptotic shape for stochastic growth models and for random clusters; existence, location and properties of phase transitions; speed of convergence to equilibrium in Markov chains, and in particular for Markov chains based on models with a phase transition; cut-off phenomena for random walks. The articles can be read independently of each other. Their unifying theme is that of models built on discrete spaces or graphs. Such models are often easy to formulate. Correspondingly, the book requires comparatively little previous knowledge of the machinery of probability.
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