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Interacting Particle Systems

Thomas M. Liggett
Springer Verlag
Publication Date: 
Number of Pages: 
Classics In Mathematics
[Reviewed by
Fernando Q. Gouvêa
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This is a paperback reprint, in Springer's Classics in Mathematics series, of a book originally published in 1985 in the Grundlehren series. The new edition is unchanged except for the addition of a six-page "Postface" and two pages of errata.

The 1986 "Telegraphic Review" described the book as "intended as reference or basis for advanced graduate course" and noted its "large bibliography" and "open problems after each chapter". It went on to say that the book "develops tools such as coupling and duality and applies these to study general spin systems, the stochastic Ising model, the voter model, the contact process, nearest particle systems, the exclusion process, and processes with unbounded values."

In the Postface, Liggett notes how the theory has changed and advanced over the last twenty years. He points out that in 1985 it was "just barely possible" to cover the existing theory in one volume; that situation has changed markedly. In particular, he remarks (with references) that about one third of the sixty-five open problems posed in the book have now been solved and (without references) that many new problems have arisen.

The reprint is well-made and easy to read. It will be welcomed by all specialists in the field.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College.

Introduction.- The Construction, and Other General Results.- Some Basic Tools.- Spin System.- Stochastic Ising Models.- The Voter Model.- The Contract Process.- Neareast-Particle Systems.- The Exclusion Process.- Linear Systems with Values in [0, infinity)s.- Bibliography.- Index.