The Self-Avoiding Walk (Probability and Its Applications) - Softcover

Madras, Neal; Slade, Gordon

Softcover

ISBN 10: 0817638911 ISBN 13: 9780817638917

Publisher: Birkhäuser, 1996

Synopsis

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. In spite of this simple definition, many of the most basic questions about this model are difficult to resolve in a mathematically rigorous fashion. In particular, we do not know much about how far an n­ step self-avoiding walk typically travels from its starting point, or even how many such walks there are. These and other important questions about the self-avoiding walk remain unsolved in the rigorous mathematical sense, although the physics and chemistry communities have reached consensus on the answers by a variety of nonrigorous methods, including computer simulations. But there has been progress among mathematicians as well, much of it in the last decade, and the primary goal of this book is to give an account of the current state of the art as far as rigorous results are concerned. A second goal of this book is to discuss some of the applications of the self-avoiding walk in physics and chemistry, and to describe some of the nonrigorous methods used in those fields. The model originated in chem­ istry several decades ago as a model for long-chain polymer molecules. Since then it has become an important model in statistical physics, as it exhibits critical behaviour analogous to that occurring in the Ising model and related systems such as percolation.

About this title

Review

"An excellent introduction for graduate students and professional probabilists... The best place to find a self-contained exposition of lace expansion."

―Bulletin of the AMS

"As a carefully written and carefully referenced exposition of an intriguing topic...this monograph is strongly recommended."

―Monatshefte Mathematik

"In this book, the reader will find basically everything there is to know about rigorous mathematical results on self-avoiding walks... It is nicely written and should be read by mathematical physicists and probabilists interested in applications to natural sciences."

―Belgian Mathematical Society

"This is the first book on self-avoiding random walk and a very good one."

―SIAM Review

"An excellent book that should be on the shelf of anyone doing work at the intersection of probability and critical phenomena... The best results about the SAW can still be found here."

--Annals of Probability

Synopsis

The self-avoiding walk is a mathematical model with important applications in statistical mechanics and polymer science. This text provides a unified account of the rigorous results for the self-avoiding walk, focusing on its critical behaviour. Topics discussed include: the lace explosion and its application to the self-avoiding walk in more than four dimensions where most issues are now resolved; an introduction to the nonrigorous scaling theory; classical work of Hammersley and others; an exposition of Kesten's pattern theorem and its consequences; a discussion of the decay of the two-point function and its relation to probabilistic renewal theory; and the role of the self-avoiding walk in physical and chemical applications.

Edition details

• Publisher: Birkhäuser
• Publication date: 1996
• Language: English
• ISBN 10: 0817638911
• ISBN 13: 9780817638917
• Binding: Paperback
• Number of pages: 441