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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. Theorem 2.1(b) of T. Hara and G. Slade, Critical behaviour of self-avoiding walk in five or more dimensions, states that for every $k\ge5$ there is a constant $D>0$ such that the mean-square displacement of the uniform $n$-step self-avoiding walk on $\mathbb{Z}^k$ is

$$\langle|\omega(n)|^2\rangle_n=Dn\,[1+O(n^{-\varepsilon})]$$

for every $\varepsilon<1/4$. Theorem 2.3 of the same announcement adds that the walk, rescaled by $n^{-1/2}$, converges in distribution to Brownian motion with diffusion constant $D$. The announcement contains no proofs; they are in the two-part paper Self-avoiding walk in five or more dimensions, Part I in Comm. Math. Phys. and Part II in Rev. Math. Phys., both linked above, which use the lace expansion with the critical bubble diagram as the small parameter and computer-assisted estimates with rigorous error bounds. By the Cauchy--Schwarz inequality the expected distance is at most the square root of the mean-square displacement, so

$$d_k(n)\le(Dn)^{1/2}(1+o(1))\ll n^{1/2}\qquad(k\ge5),$$

a remark of this page, as on the [[../library/discrete_geometry/hara_1991_critical_behaviour_self_avoiding_walk_five_more_dimensions/_index|source card]], not of the paper. This answers the problem's second question yes for every $k\ge5$, extending Slade's theorem for all sufficiently large $k$ to an explicit threshold. The site's commentary states the asymptotic $d_k(n)\sim Dn^{1/2}$ for the expected distance itself; only the upper bound is claimed here.

Covers. The second question for every $k\ge5$, answered yes. Not covered: $k=3$ and $k=4$, where the site records the conjecture that the bound is false, and the first question, on the plane.

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

Acceptance. Refereed: the announcement, T. Hara and G. Slade, Critical behaviour of self-avoiding walk in five or more dimensions, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 417--423, received 30 January 1991; the proofs, Self-avoiding walk in five or more dimensions. I. The critical behaviour, Comm. Math. Phys. 147 (1992), no. 1, 101--136, and The lace expansion for self-avoiding walk in five or more dimensions, Rev. Math. Phys. 4 (1992), no. 2, 235--327. The site's commentary records the result for $k\ge5$, but the site labels the problem OPEN, so that remark is not acceptance of the problem and the page lists no reviewed evidence. The proofs are not compiled in this corpus.

Dating. The page is dated by the issue month of the announcement in the publisher's record, October 1991; the day is a placeholder. The two proof papers appeared in June 1992.