Self-Avoiding Walk

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.

"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