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 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!

verygentle introduction. This is about the extent of my knowledge though... $\endgroup$Introduction to Polymer Physicsby Doi. $\endgroup$J. Statist. Phys.,J. Stat. Mech. Theory Exp.,J. Math Phys.,Phys. Let., etc. I may be in over my head, but it should be a fun swim! $\endgroup$1more comment