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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!

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Random knots :) –  Tom Boardman Jul 2 '10 at 10:31
    
See this old blog post: maa.org/mathland/mathtrek_11_3.html for a very gentle introduction. This is about the extent of my knowledge though... –  Tom Boardman Jul 2 '10 at 10:36
    
@Tom Boardman: Yes, that work of Sumners and Whittington on random knots is fascinating! Whether knot theory is analogous to diffusion in this situation is not clear to me. I am seeking a continuous theory that parallels and informs the discrete. –  Joseph O'Rourke Jul 2 '10 at 12:01
    
This might be off-topic for this site, but you may also want to look at polymer physics books to see how physicists treat problems involving lots of long chains of molecules. (Surprisingly often, self-avoidance turns out to be unimportant to the properties of polymeric materials.) There's an article by Tom McLeish in a recent Physics Today that serves as a teaser to the field dx.doi.org/10.1063/1.2970211 (it may be easier to find if you go to your institution's physics library and browse through recent issues), and a good introductory book is Introduction to Polymer Physics by Doi. –  j.c. Jul 2 '10 at 13:05
    
@jc: Thanks for the connection. I can see that many of the references in the Lawler paper to which Yuri pointed me are to journals such as 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! –  Joseph O'Rourke Jul 2 '10 at 13:17

2 Answers 2

up vote 12 down vote accepted

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.

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SLE = ? Sorry for my ignorance... –  Joseph O'Rourke Jul 2 '10 at 12:26
    
SLE=Schramm–Loewner evolution, see, e.g., arxiv.org/abs/0712.3256 –  Yuri Bakhtin Jul 2 '10 at 12:40
    
Thank you! [Interesting: "Thank you!" is too short a comment for MO! So I had to add this pointless parenthetical remark to post.] –  Joseph O'Rourke Jul 2 '10 at 12:47
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Well, and in dimensions $5$ and higher, the scaling limit is again Brownian motion, as proved long ago by Hara and Slade. –  Yvan Velenik Jul 2 '10 at 13:14
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@Joseph and @Wadim, re short comments: mathoverflow.net/questions/18263/… –  Mark Meckes Jul 2 '10 at 13:49

For SLE see http://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution

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Ah, thank you!! It may be that Schramm–Loewner evolution is the theory I seek... –  Joseph O'Rourke Jul 2 '10 at 12:40
    
@Joseph: "SLE" could be too short for an answer! :-) –  Wadim Zudilin Jul 2 '10 at 12:57

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