Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 7

Problem 528

claims/: The 4 claim pages of Problem 528, one per claimant's result; the problem's standing derives from them.

Statement. Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine

C_k=\lim_{n\to\infty}f(n,k)^{1/n}.

Status. Open, in the site's label. The accepted partial claims Kesten 1964 and [[problems/discrete_geometry/E0528/claims/2007_08_21_clisby_liang_slade|Clisby, Liang and Slade 2007]] give the asymptotics of $C_k$ in powers of $1/k$ . [[problems/discrete_geometry/E0528/claims/1993_08_07_conway_guttmann|Conway and Guttmann 1993]] and Alm 1993 bound $2.62\le C_2\le2.696$ . None determines $C_k$ for any $k\ge2$ . Hammersley and Morton [HM54] prove only that the limit exists, and the value of $C_2$ in [JSG16] is a numerical estimate, not a proof, so neither has a claim page.

Source. erdosproblems.com/528, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #528, https://www.erdosproblems.com/528.

References.

[Al93] Alm, Sven Erick, Upper bounds for the connective constant of self-avoiding walks. Combin. Probab. Comput. (1993), 115-136.
[CG93] Conway, A. R. and Guttmann, A. J., Lower bound on the connective constant for square lattice self-avoiding walks. J. Phys. A (1993), 3719-3724.
[CLS07] Clisby, Nathan and Liang, Richard and Slade, Gordon, Self-avoiding walk enumeration via the lace expansion. J. Phys. A (2007), 10973-11017.
[HM54] Hammersley, J. M. and Morton, K. W., Poor man's Monte Carlo. J. Roy. Statist. Soc. Ser. B (1954), 23-38; discussion 61-75.
[JSG16] Jacobsen, Jesper Lykke and Scullard, Christian R. and Guttmann, Anthony J., On the growth constant for square-lattice self-avoiding walks. J. Phys. A (2016), 494004, 18.
[Ke63] Kesten, Harry, On the number of self-avoiding walks. J. Mathematical Phys. (1963), 960-969. This paper proves the ratio limit of the walk counts, not an expansion of $C_k$ . The site's commentary credits Kesten under this key with $C_k=2k-1-1/(2k)+O(1/k^2)$ ; that expansion is proved in the sequel, H. Kesten, On the number of self-avoiding walks. II, J. Math. Phys. 5 (1964), no. 8, 1128-1137, DOI 10.1063/1.1704216, to which Clisby, Liang and Slade [CLS07] attribute it.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (12)

Kesten's expansion of the connective constant in powers of 1/k Alm's upper bound 2.696 for the square lattice Conway and Guttmann's lower bound 2.62 for the square lattice Clisby, Liang and Slade's 1/(2k) expansion of the connective constant problems/discrete_geometry/E0528/claims Discrete and Convex Geometry discrete_geometry/clisby_2007_self_avoiding_walk_enumeration_via_lace Section 1.2: exact counts of self-avoiding walks and polygons in dimensions d >= 3 Equation (1): the 1/d expansion of the connective constant through order (2d)^{-11}Section 7.2: rigorous upper bounds on the connective constant for 3 <= d <= 12 discrete_geometry/jacobsen_2016_growth_constant_square_lattice_self_avoiding Principal result (pp. 1, 22, unnumbered): the numerical estimate mu = 2.63815853032790(3) for square-lattice self-avoiding walks

Graph