Problem 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 $k\geq 3$ ?

Source: erdosproblems.com/529

Geometry
Probability

Wiki page

Status

Open

No claim settles this problem.

Open, in the site's label. The accepted partial claims Slade 1987 and Hara and Slade 1991 answer the second question yes for all sufficiently large $k$ and for every $k\ge5$ . No claim covers $k=3$ or $k=4$ or the first question. Duminil-Copin and Hammond's $d_2(n)=o(n)$ [DuHa13] settles neither question and has no claim page.

DuHa13: Duminil-Copin, Hugo and Hammond, Alan, Self-avoiding walk is sub-ballistic. Comm. Math. Phys. (2013), 401-423.
HaSl91: Hara, Takashi and Slade, Gordon, Critical behaviour of self-avoiding walk in five or more dimensions. Bull. Amer. Math. Soc. (N.S.) (1991), 417-423.
HaSl92: Hara, Takashi and Slade, Gordon, Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. (1992), 101-136.
MaSl93: Madras, Neal and Slade, Gordon, The self-avoiding walk. (1993), xiv+425.
Sl87: Slade, Gordon, The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. (1987), 661-683.

Problem 529

Status

Proof claims

Discussion