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Preliminary:

Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks like a $d$ dimensional Gaussian.

A self-avoiding walk (SAW), looks to be well studied in Mathematical and Physics literature. I've found numerous examples of average properties of these walks. Average end-to-end distance, average radius of gyration, average diameter, etc... I'd like to know about the PDF of the final position of a SAW, something analogous to the fact that a normal random walk looks like a Gaussian.

Empirical data:

It was easy to code up some simulation data. I took $8*10^7$ samples of a SAW for $N=26$ in $d=2$. The resulting PDF is the top graph. The empty squares are due to the parity of the walk (some sites are even/odd sites that are only accessible when $N$ is even/odd) What I'm really interested in is the bottom graph, where I took the average value of the PDF for a given radius $P_{SAW}(r; d)$:

enter image description here

For comparisons sake, the same data was computed for a regular random walk:

enter image description here

Question:

What can we say about $P_{SAW}(r; d)$?

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    @NateEldredge If you want to turn that into an answer I'll accept it (perhaps with a small description of SLE_{8/3}). I didn't not realize that this was an open question.2012-08-01

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