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Claim. Let $C_k$ be the connective constant of Problem 528, written $\mu$ in the paper with $d$ for the dimension $k$. N. Clisby, R. Liang and G. Slade, Self-avoiding walk enumeration via the lace expansion, extend the known expansion of $C_k$ in powers of $1/(2k)$, whose first terms are Kesten's $2k-1-1/(2k)$, through the term of order $(2k)^{-11}$, with a remainder $O((2k)^{-12})$. Section 1.3 of the paper states that this error estimate is rigorous; it rests on the proof, which the paper cites, that the expansion exists to all orders, and the new coefficients are computed from the exact enumeration of walks and polygons in every dimension that the paper's lace-expansion method and two-step algorithm produce (24-step walks and polygons in all $k\ge4$, longer series in $k=3$ and $k=4$). The coefficients are not transcribed on this page. The [[../library/discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/_index|source card]] records the enumeration results and the paper's edition.

Covers. The asymptotic expansion of $C_k$ in powers of $1/(2k)$ through order $(2k)^{-11}$ with its error term. Not covered, and not claimed: the paper's series-analysis estimates of $C_k$ and of critical exponents and amplitudes for $3\le k\le8$, which are numerical estimates, not proofs; and the value of $C_k$ for any fixed $k\ge2$, which the expansion does not determine.

Depends on. Nothing in this wiki; the claim rests on the cited paper and the earlier work it cites for the existence of the expansion.

Acceptance. Refereed: N. Clisby, R. Liang and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A 40 (2007), no. 36, 10973--11017. The site's commentary records the paper as giving more precise asymptotics than Kesten's, 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 article's online publication date in the publisher's record, 21 August 2007; no earlier public posting is recorded.