The Self-Avoiding Walk

  • Neal Madras
  • Gordon Slade

Part of the Probability and Its Applications book series (PA)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Neal Madras, Gordon Slade
    Pages 1-33
  3. Neal Madras, Gordon Slade
    Pages 35-55
  4. Neal Madras, Gordon Slade
    Pages 57-76
  5. Neal Madras, Gordon Slade
    Pages 77-117
  6. Neal Madras, Gordon Slade
    Pages 119-169
  7. Neal Madras, Gordon Slade
    Pages 171-228
  8. Neal Madras, Gordon Slade
    Pages 229-255
  9. Neal Madras, Gordon Slade
    Pages 257-279
  10. Neal Madras, Gordon Slade
    Pages 281-364
  11. Neal Madras, Gordon Slade
    Pages 365-374
  12. Back Matter
    Pages 375-425

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • Neal Madras
    • 1
  • Gordon Slade
    • 2
  1. 1.Department of Mathematics and StatisticsYork UniversityDownsviewCanada
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-4132-4
  • Copyright Information Birkhäuser Boston 1996
  • Publisher Name Birkhäuser Boston
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-8176-3891-7
  • Online ISBN 978-1-4612-4132-4
  • About this book