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."
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