One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a natural continuous analog of discrete self-avoiding walks?
I am particularly interested in self-avoiding polygons,
i.e., closed self-avoiding walks.


I've only found reference
(in Madras and Slade,
*The Self-Avoiding Walk*, p.365ff)
to continuous analogs of "weakly self-avoiding walks"
(or "self-repellent walks") which discourage but
do not forbid self-intersection.


I realize this is a vague question,
reflecting my ignorance of the topic.
But perhaps those knowledgeable could point me to the
the right concepts. Thanks!


**Addendum**.
[Schramm–Loewner evolution](https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution) is the answer. It is the conjectured scaling limit of the self-avoiding walk and several other related stochastic processes. Conjectured in low dimensions,
proved in high dimensions, as pointed out by Yuri and Yvan. Many thanks for your help!



Random walk is to diffusion as self-avoiding random walk is to ...?
In 2D the scaling limit is believed to be SLE with parameter 8/3. This was conjectured by Lawler, Schramm and Werner and, to the best of my knowledge, still remains open.


