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Statement

Notation (p. 3). $c_n$ is the number of $n$-step self-avoiding walks on $\mathbb{Z}^d$ starting at $0$, and $\mu=\lim_{n\to\infty}c_n^{1/n}$ is the connective constant; the limit exists by subadditivity (Hammersley and Morton, the paper's [29]). Hara and Slade (the paper's [32]) proved that $\mu$ has an asymptotic expansion in powers of $1/d$ to all orders, with integer coefficients.

Equation (1) (p. 3). As $d\to\infty$,

$$\begin{aligned} \mu={}&2d-1-\frac{1}{2d}-\frac{3}{(2d)^2}-\frac{16}{(2d)^3}-\frac{102}{(2d)^4}-\frac{729}{(2d)^5}-\frac{5533}{(2d)^6}-\frac{42229}{(2d)^7}\\ &-\frac{288761}{(2d)^8}-\frac{1026328}{(2d)^9}+\frac{21070667}{(2d)^{10}}+\frac{780280468}{(2d)^{11}}+O\left(\frac{1}{(2d)^{12}}\right). \end{aligned}$$

What the paper says of it (p. 4). Kesten (the paper's [42], 1964) proved $\mu=2d-1-\frac{1}{2d}+O((2d)^{-2})$. The coefficients through the term $-102/(2d)^4$ were known before (the paper's [14, 32, 52], with a rigorous error estimate in [32]; the print writes this term as "$102(2d)^{-5}$" [sic]); the other seven are new, and the paper states that the error estimate in (1) is rigorous. The paper remarks that the series appears to have radius of convergence zero but that it has no proof of this, and notes the change of sign at the term of order $(2d)^{-10}$.

The intermediate expansion (39) (p. 19) gives $z_c=1/\mu$ in powers of $1/(2d)$ through order $(2d)^{-13}$ with error $O((2d)^{-14})$; (1) is its reciprocal.

Source. Nathan Clisby, Richard Liang and Gordon Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. 40 (2007), 10973-11017, DOI 10.1088/1751-8113/40/36/003. Pages are those of the authors' manuscript dated July 24, 2007, the edition identified on the [[discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/_index|source card]]: equation (1) on p. 3, the commentary on p. 4, the derivation in Section 4.1 on p. 19, the error bounds (35)-(36) on pp. 18-19 and the proof of (36) in Section 4.3, pp. 20-23.

Read depth. Claims checked: equation (1), its notation and the commentary were read clause by clause on the printed pages, and the coefficients were compared with the print. The proof of (36) (pp. 20-23) was read but not checked step by step, and the coefficients rest on the paper's computer enumeration, which was not reproduced. Nothing here is independently reviewed.

Proof pointer

Section 4.1 (p. 19). For $d\ge5$, Hara and Slade's lace-expansion identity (37) (the paper's [31]) writes $z_c=\frac{1}{2d}\bigl(1-\sum_{m\ge2}\pi_m z_c^m\bigr)$, where $\pi_m=\sum_{N}(-1)^N\pi_m^{(N)}$ counts lace graphs. The standard estimate (35), that the terms with $N$ or more laces contribute $O(d^{-N})$, together with the new estimate (36), that the terms with $m\ge j$ steps contribute $O(d^{-j/2})$ (proved in Section 4.3), truncates (37) to the finitely many $\pi_m^{(M)}$ with $m\le2N$ and $M\le N$, giving (38) with error $O(d^{-N-2})$; the existence of the all-order expansion from [32] removes a fractional power from the error. The counts $\pi_m^{(M)}$ for $m\le24$ and $M\le12$, which are polynomials in $d$ by the decomposition by dimension (31) (p. 15), are fed into (38) recursively to give (39), and inverting gives (1).

Dependencies

The paper's lace-expansion enumeration of $\pi_{m,\delta}^{(N)}$, recorded on [[discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace/enumeration_results|the enumeration results page]]; the external results (37) and (35), from Hara and Slade [31] and Slade's lecture notes [59]; the existence of the all-order expansion, from Hara and Slade [32].

Bears on

• Problem 528: the problem's $C_k$ is the paper's $\mu$ with $d=k$, and equation (1) gives the asymptotics of $C_k$ as $k\to\infty$ through the term of order $(2k)^{-11}$, with remainder $O((2k)^{-12})$. It does not determine $C_k$ for any fixed $k$. The problem's claim page [[../wiki/problems/discrete_geometry/E0528/claims/2007_08_21_clisby_liang_slade|for this paper]] records this expansion.