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 $P_{SAW}(r; d)$:

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

Question:

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

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It's not that easy to select a SAW uniformly at random from all those with a particular length. How did you go about it? – mjqxxxx Jul 31 '12 at 17:14
@mjqxxxx Correct: if you do it naively, you have to reject almost all samples for $d=2$ as the chance of overlap is high. I used the Rosenbluth-Roesenbluth method (RR) I found in a paper that selects from moves available as the chain grows and weights them accordingly. dx.doi.org/10.1088/0305-4470/21/1/020 – Hooked Jul 31 '12 at 17:45
check the (brief) discussion here: arxiv.org/pdf/math/0602151.pdf – mike Jul 31 '12 at 18:03
+1 to Mike's comment; see section 2.2. It is a famous open conjecture that the scaling limit of SAW is $SLE_{8/3}$, which I think would address your question. You are actually not asking for the distribution of the limit process but only its endpoint distribution, which could in principle be easier, though I would guess not. – Nate Eldredge Aug 1 '12 at 14:28
@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. – Hooked Aug 1 '12 at 14:35