Let $\gamma(i)$ be a self avoiding walk (SAW) on a 2D lattice $L$ (a square lattice for example) starting at a predefined origin ( $\gamma(0)=(0,0)$ ) and having length $n:=\ell(\gamma)$. Furthermore, suppose $\gamma$ ends at a point $u$, so that $\gamma(n)=u$. Let $N(u)$ be the set of neighboring vertices of $u$ (that are connected to $u$). If I were to further condition the SAW by clamping the second-to-last-step, i.e. $\gamma(n-1)=v$, what can be said about the probability that the walk visited any of the other neighbors of $u$. Naturally, the length of the SAW and the distance of $u$ from the origin will be interlocked in a tug of war. I am primarily interested in seeing when the probability of visiting adjacent neighbors tends to zero. 



Self Avoiding Walk Pair Correlation
If the walks are sampled uniformly at random, then the probability of another neighbour of $u$ being visited is related to the "atmosphere" of a walk, and the connective constant $\mu$. The mean number of additional neighbours approaches $3-\mu \approx 0.361841469\cdots$ for SAWs on $Z^2$, while the probability of there being at least one additional neighbour approaches $1-0.71114 = 0.28886$ (from eq. 12, Owczarek and Prellberg, ref. below).


A recent article gave probabilities for having 0, 1, 2, and 3 unoccupied neighbours, A L Owczarek and T Prellberg, "Scaling of the atmosphere of self-avoiding walks", J. Phys. A: Math. Theor. 41 (2008) 375004 (6pp).


The most accurate estimate of $\mu$ for $Z^2$ comes from self-avoiding polygon enumerations by Iwan Jensen, "A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice", J. Phys. A: Math. Gen. 36 (2003) 5731–5745, also on the arXiv: <http://arXiv.org/abs/cond-mat/0301468v1>


If you're wondering about the scaling behaviour as $|u|$ is varied, then I think that given $x = |u|/\langle R\_e^2 \rangle^{1/2}$, the scaling function $P(x)$ for the probability of more than one neighbour being occupied must go to zero as $x \rightarrow \infty$, but I don't have any intuition yet as to how it will approach zero (I think this is a nice question).


