---

Source. Jesper Lykke Jacobsen, Christian R. Scullard and Anthony J. Guttmann, On the growth constant for square-lattice self-avoiding walks, J. Phys. A 49 (2016), no. 49, 494004, DOI 10.1088/1751-8113/49/49/494004: the abstract (p. 1), Section 4.5 (pp. 20--22) and the Conclusion (pp. 22--23), with the conjecture in Section 1 (pp. 1--3). Pages are those of the arXiv version arXiv:1607.02984v1, as identified on the source card.

Statement

The paper has no numbered theorems. Its principal result, so called in the Conclusion (p. 22), is a numerical estimate, not a theorem: the parenthesized digit is an error bar from the authors' extrapolation, not a proved bound.

Setting. $\mu$ is the growth constant of self-avoiding walks on the square lattice, and $x_c=1/\mu$ is the radius of convergence of their generating function (p. 2). For each circumference $n$ the topological transfer-matrix method gives a finite-size value $x_c(n)$; Table 4 (p. 21) lists these to 40 digits for $2\le n\le21$.

Estimate (p. 22, eq. (25), and Conclusion, p. 22). Assuming that $x_c(n)$ has the power-law scaling form (21), $x_c(n)=x_c+\sum_{k\ge1}A_k/n^{\Delta_k}$, and taking the exponents to be $\Delta_k=2(k+1)$ for every $k\ge1$ (eq. (22), p. 21), the authors fit the data to obtain

$$x_c=0.379052277755161\,(5),\qquad \mu=2.63815853032790\,(3).$$

The exponents $\Delta_1=4$ and $\Delta_2=6$ are fitted from the data ($\Delta_1=4.000\,000(1)$, $\Delta_2=6.000(4)$, p. 20); the general form (22) is an assumption the paper states and then uses (p. 21). The final value and error bar come from comparing the fits of orders $8\le M\le16$ (p. 22). The authors report that repeating the whole procedure with $n_{\max}=20$ and $n_{\max}=19$ gives compatible, less accurate results (p. 22).

The conjecture (pp. 1--3, 21). Guttmann's earlier conjecture takes $\mu$ to be the positive real root $t=2.6381585303417408684303\cdots$ of $13t^4-7t^2-581$ (eq. (1), p. 2), so that $x_c^{\mathrm{conj}}=1/\mu=0.37905227775317290937028\cdots$ (eq. (2), p. 2). The abstract says the conjecture "fails in the 12th digit" (p. 1, quoted), and the introduction that (2) "is too low by about $2\cdot10^{-12}$" (p. 3, quoted). The paper also notes (p. 2) that the quartic has, besides its root at $-2.6381585303417408684303\cdots$, a conjugate pair of roots on the imaginary axis, and that numerical analysis of the walk and polygon generating functions has shown no such singularity.

Earlier estimates the paper compares (pp. 2, 7, 12). The polygon-series estimate of Clisby and Jensen, $\mu=2.63815853035(2)$, that is $x_c=0.379052277752(3)$ (eq. (4), p. 7), and the paper's own estimate by the adapted Duminil-Copin and Smirnov identity, $x_c=0.379052277750\pm0.0000000005$ (eq. (17), p. 12), both agree with (2) within their stated uncertainty.

Consistency checks (observations of this page, not of the paper). The reciprocal of $0.379052277755161$ is $2.638158530327904\ldots$, and an error of $5\cdot10^{-15}$ in $x_c$ corresponds to about $3.5\cdot10^{-14}$ in $\mu$, matching the stated $\mu$ and its error bar. The positive root of $13t^4-7t^2-581$ has reciprocal $0.379052277753172\ldots$, which is $1.99\cdot10^{-12}$ below the value (25). The paper's $\mu$, the limit of $c_n^{1/n}$ for the number $c_n$ of $n$-step self-avoiding walks from the origin of $\mathbb Z^2$, is the quantity $C_2$ of Problem 528.

Read depth. Claims checked: the estimate, the scaling assumptions, the conjecture and the comparisons were read clause by clause on the printed pages. The transfer-matrix computation and the extrapolation were read for their structure only and not rerun.

Proof pointer

There is no proof; the result is numerical. Section 4 (pp. 13--22) describes the topological transfer-matrix method, which locates $x_c(n)$ by equating the leading eigenvalues of the transfer matrix of a semi-infinite cylinder of circumference $n$ in two topological sectors (p. 14), its parallel implementation up to $n=21$, and the two-step extrapolation (23)--(24) of Section 4.5 (pp. 20--22).

Dependencies

The scaling form (21) with the exponent assumption (22), and the computed values $x_c(n)$ of Table 4 (p. 21), of which those with $n\le18$ and the first 22 digits of $n=19$ appeared in earlier work of Jacobsen alone (caption of Table 4, p. 21; see also p. 14).

Bears on

• Problem 528: the paper gives a numerical estimate of $C_2$, with its error bar in the fourteenth decimal place, and its authors conclude that $C_2$ is not the positive root of $13t^4-7t^2-581$. Neither statement is proved: the estimate rests on an extrapolation under the assumed scaling form, so the paper does not determine $C_2$, bound it rigorously, or prove that this root is not its value. It says nothing about $C_k$ for $k\ne2$.