---

Claim. Let $C_k$ be the connective constant of Problem 528, the limit of $f(n,k)^{1/n}$ where $f(n,k)$ counts the $n$-step self-avoiding walks from the origin in $\mathbb{Z}^k$. The theorem of H. Kesten, On the number of self-avoiding walks. II, states that

$$C_k=2k-1-\frac{1}{2k}+O\!\left(\frac{1}{k^2}\right)\qquad(k\to\infty).$$

The paper compares $C_k$ with the growth constant $\beta_{k,2r}$ of the walks that may revisit a point only after more than $2r$ steps, proves $\beta_{k,2r}-C_k=O(k^{-r})$ as $k\to\infty$, and reads the expansion off this comparison; the abstract states both results in this form. The site's commentary credits the expansion to Kesten under its key [Ke63], which resolves to the first paper of the series (J. Math. Phys. 4 (1963), 960--969); that paper proves the ratio limit of the walk counts, and Clisby, Liang and Slade, whose [[problems/discrete_geometry/E0528/claims/2007_08_21_clisby_liang_slade|later expansion]] extends this one, cite the 1964 sequel for it. The problem page annotates the key.

Covers. The first terms of the asymptotic expansion of $C_k$ as $k\to\infty$: the leading term $2k-1$, the correction $-1/(2k)$ and the order $O(k^{-2})$ of the remainder. Not covered: the value of $C_k$ for any fixed $k\ge2$; no $C_k$ is determined, and the problem's question is not answered.

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

Acceptance. Refereed: H. Kesten, On the number of self-avoiding walks. II, J. Math. Phys. 5 (1964), no. 8, 1128--1137. The site's commentary records the expansion, 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, August 1964; the day is a placeholder.