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 justifi cation for going to this level of generality.

Percolation theory is the study of an idealized random medium in two or more dimensions. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. Much new material appears in this second edition including dynamic and static renormalization, strict inequalities between critical points, a sketch of the lace expansion, and several essays on related fields and applications.

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.

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

This book, first published in 2006, is an account of percolation theory and its ramifications.

This work dealing with percolation theory clustering, criticallity, diffusion, fractals and phase transitions takes a broad approach to the subject, covering basic theory and also specialized fields like disordered systems and renormalization groups.

This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.