Erdős Problem 529

### What is the conjecture

https://www.erdosproblems.com/529

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 $k\geq 3$?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else