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on Erdős #529 · geometry / probability

The statement

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 $k\geq 3$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.