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. This model is an example of a logarithmic CFT, which are non-unitarity.

• Wikipedia
• seeCardy's notes on SAWs from a physics point of view
• see 1206.2092 for lectures on SAWs from a mathematical viewpoint
• see 1605.03959 on logarithmic CFTs for a discussion on the $O(0)$ model