A self-avoiding random walk (or self-avoiding walk, or SAW) is a path on a $d$-dimensional lattice in $\mathbb{Z}^d$ which never visits the same point more than once. As a statistical model, they are defined as the set of all such paths with length $n$ weighted with equal probability. A number of questions concerning their behavior as $n \to \infty$ are still open, and the models display a rich variety of critical behavior.

This is the $n\rightarrow 0$ specialization of the $O(N)$ model.

• Wikipedia
• see 1206.2092 for a nice set of introductory lectures
• see 1605.03959 on Logarithmic CFTs for a discussion on the $O(0)$ model