The Self-Avoiding Walk

1. Front Matter

   Pages i-xiv

   PDF

2. Introduction

   • Neal Madras, Gordon Slade

   Pages 1-33

3. Scaling, polymers and spins

   • Neal Madras, Gordon Slade

   Pages 35-55

4. Some combinatorial bounds

   • Neal Madras, Gordon Slade

   Pages 57-76

5. Decay of the two-point function

   • Neal Madras, Gordon Slade

   Pages 77-117

6. The lace expansion

   • Neal Madras, Gordon Slade

   Pages 119-169

7. Above four dimensions

   • Neal Madras, Gordon Slade

   Pages 171-228

8. Pattern theorems

   • Neal Madras, Gordon Slade

   Pages 229-255

9. Polygons, slabs, bridges and knots

   • Neal Madras, Gordon Slade

   Pages 257-279

10. Analysis of Monte Carlo methods

    • Neal Madras, Gordon Slade

    Pages 281-364

11. Related topics

    • Neal Madras, Gordon Slade

    Pages 365-374

12. Back Matter

    Pages 375-425

    PDF

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.

• Probability
• Walk
• algorithms
• Brownian motion
• computer simulation
• finite-dimensional distribution
• loop-erased random walk
• Mathematica
• Monte Carlo method
• probability
• Random Walk
• renewal theory
• statistical physics

"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

• Department of Mathematics and Statistics, York University, Downsview, Canada

  Neal Madras

• Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

  Gordon Slade

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