OPEN
Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self intersections) - that is, a
self-avoiding walk. Is it true that
\[\lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty?\]
Is it true that
\[d_k(n)\ll n^{1/2}\]
for $k\geq 3$?
Slade
[Sl87] proved that, for $k$ sufficiently large, $d_k(n)\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade (
[HASl91] and
[HaSl92]) proved this for all $k\geq 5$.
For $k=2$ Duminil-Copin and Hammond [DuHa13] have proved that $d_2(n)=o(n)$.
It is now conjectured that $d_k(n)\ll n^{1/2}$ is false for $k=3$ and $k=4$, and more precisely (see for example Section 1.4 of [MaSl93]) that $d_2(n)\sim Dn^{3/4}$, $d_3(n)\sim n^{\nu}$ where $\nu\approx 0.59$, and $d_4(n)\sim D(\log n)^{1/8}n^{1/2}$.
Madras and Slade [MaSl93] have a monograph on the topic of self-avoiding walks.
See also [528].
