Problem 529

Erdős Problem #529: declared status 'open'. Formalized: no. 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 $k\geq 3$? Current best: Hara and Slade ([HaSl91] and [HaSl92]) proved this for all $k\geq 5$. Prize: no. Tags: geometry, probability.