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Claim. Let $d_k(n)$ be the expected distance from the origin of the uniform $n$-step self-avoiding walk on $\mathbb{Z}^k$, as in Problem 529. G. Slade, The diffusion of self-avoiding random walk in high dimensions, proves that there is a dimension $k_0$ such that for every $k\ge k_0$ the mean-square displacement of the uniform $n$-step self-avoiding walk on $\mathbb{Z}^k$ is asymptotic to $Dn$ as $n\to\infty$, for a constant $D>0$; the proof uses the lace expansion of Brydges and Spencer. This is the theorem as the zbMATH review of the paper (Zbl 0628.60073) states it; no explicit value of $k_0$ is given. By the Cauchy--Schwarz inequality the expected distance is at most the square root of the mean-square displacement, so

$$d_k(n)\le(Dn)^{1/2}(1+o(1))\ll n^{1/2}\qquad(k\ge k_0),$$

a remark of this page, not of the paper. This answers the problem's second question yes for all sufficiently large $k$. The threshold was later brought down to every $k\ge5$ by [[problems/discrete_geometry/E0529/claims/1991_10_01_hara_slade|Hara and Slade]]. The site's commentary states the stronger asymptotic $d_k(n)\sim Dn^{1/2}$ for the expected distance itself; the review states the theorem for the mean-square displacement, and only the upper bound is claimed here.

Covers. The second question for all sufficiently large $k$, answered yes with no explicit threshold. Not covered: $k=3$, $k=4$, any explicit dimension, and the first question, on the plane.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: G. Slade, The diffusion of self-avoiding random walk in high dimensions, Comm. Math. Phys. 110 (1987), no. 4, 661--683. The site's commentary records the result, but the site labels the problem OPEN, so that remark is not acceptance of the problem and the page lists no reviewed evidence. The proof is not compiled in this corpus.

Dating. The page is dated by the issue month in the publisher's record, December 1987; the day is a placeholder.