Problem 529

../

claims/: The 2 claim pages of Problem 529, one per claimant's result; the problem's standing derives from them.

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Statement. Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that

$$\lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty?$$

Is it true that

$$d_k(n)\ll n^{1/2}$$

for $k\geq 3$?

Status. Open, in the site's label. The accepted partial claims Slade 1987 and [[problems/discrete_geometry/E0529/claims/1991_10_01_hara_slade|Hara and Slade 1991]] answer the second question yes for all sufficiently large $k$ and for every $k\ge5$. No claim covers $k=3$ or $k=4$ or the first question. Duminil-Copin and Hammond's $d_2(n)=o(n)$ [DuHa13] settles neither question and has no claim page.

Source. erdosproblems.com/529, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #529, https://www.erdosproblems.com/529.

References.

• [DuHa13] Duminil-Copin, Hugo and Hammond, Alan, Self-avoiding walk is sub-ballistic. Comm. Math. Phys. (2013), 401-423.
• [HaSl91] Hara, Takashi and Slade, Gordon, Critical behaviour of self-avoiding walk in five or more dimensions. Bull. Amer. Math. Soc. (N.S.) (1991), 417-423.
• [HaSl92] Hara, Takashi and Slade, Gordon, Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. (1992), 101-136.
• [MaSl93] Madras, Neal and Slade, Gordon, The self-avoiding walk. (1993), xiv+425.
• [Sl87] Slade, Gordon, The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. (1987), 661-683.

Formalization. None recorded.

Current assessment

The second question is answered yes in every dimension $k\ge5$: Slade [Sl87] proved that for all sufficiently large $k$ the mean-square displacement of the uniform $n$-step self-avoiding walk on $\mathbb{Z}^k$ is asymptotic to $Dn$, and Hara and Slade [HaSl91, HaSl92] proved $Dn\,[1+O(n^{-\varepsilon})]$ for every $k\ge5$ and every $\varepsilon<1/4$, with a Brownian scaling limit; the Cauchy--Schwarz inequality turns each into $d_k(n)\ll n^{1/2}$. The claim pages Slade 1987 and [[problems/discrete_geometry/E0529/claims/1991_10_01_hara_slade|Hara and Slade 1991]] record these as accepted partial claims on their refereed publication. For $k=3$ and $k=4$ the site's commentary records the conjecture that $d_k(n)\ll n^{1/2}$ is false, with the predicted asymptotics $d_3(n)\sim n^{\nu}$, $\nu\approx0.59$, and $d_4(n)\sim D(\log n)^{1/8}n^{1/2}$ (Section 1.4 of [MaSl93]); nothing is proved there. The first question is open: the predicted $d_2(n)\sim Dn^{3/4}$ is unproved, and Duminil-Copin and Hammond [DuHa13] prove only that the walk is sub-ballistic, $d_2(n)=o(n)$, which bounds $d_2(n)$ from above and settles neither question, so it has no claim page.

No claim page records family 237 of the OpenAI mathematics release, which names this problem. Two of its manuscripts claim fixed-length laws for uniform self-avoiding walks on the honeycomb lattice. [[../library/discrete_geometry/openai_2026_mass_covering_exponents_fixed_length_honeycomb_walks/_index|Mass and covering exponents for fixed-length honeycomb walks]] claims diameter $n^{3/4+o(1)}$ outside an event of polynomially small probability at every large length (its Theorem 1.1 and Corollary 1.4). [[../library/discrete_geometry/openai_2026_renewal_changes_law_critical_honeycomb_walks/_index|Renewal and changes of law for critical honeycomb walks]] claims endpoint distance $n^{3/4+o(1)}$ in probability, with matching moments, along a set of lengths of natural density one (its Theorem 8.2). The third, [[../library/discrete_geometry/openai_2026_critical_strip_crossing_mass_honeycomb_lattice/_index|Critical strip-crossing mass on the honeycomb lattice]], supplies strip-crossing mass and displacement exponents and claims no fixed-length law. Together they would give the honeycomb analogue of the first question along a density-one set of lengths, not at every length. The problem is posed on $\mathbb{Z}^k$. The manuscripts' analytic inputs come from an observable special to the honeycomb lattice, and they claim no transfer between lattices, so they settle no instance of either question. The release's Lean covers only supporting statements (bridge finiteness, the free-energy limit and the strip-crossing mass), not the $3/4$ laws. The release's README states that its manuscripts were produced by an internal OpenAI model at different stages of verification.

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.

• duminilcopin_2013_self_avoiding_walk_is_sub_ballistic
• duminilcopin_2013_self_avoiding_walk_is_sub_ballistic / corollary_1_2
• duminilcopin_2013_self_avoiding_walk_is_sub_ballistic / corollary_1_3
• duminilcopin_2013_self_avoiding_walk_is_sub_ballistic / theorem_1_1
• hara_1991_critical_behaviour_self_avoiding_walk_five_more_dimensions
• hara_1991_critical_behaviour_self_avoiding_walk_five_more_dimensions / theorem_2_1
• hara_1991_critical_behaviour_self_avoiding_walk_five_more_dimensions / theorem_2_3
• openai_2026_critical_strip_crossing_mass_honeycomb_lattice
• openai_2026_critical_strip_crossing_mass_honeycomb_lattice / theorem_1_1
• openai_2026_mass_covering_exponents_fixed_length_honeycomb_walks
• openai_2026_mass_covering_exponents_fixed_length_honeycomb_walks / corollary_1_4
• openai_2026_mass_covering_exponents_fixed_length_honeycomb_walks / theorem_1_1
• openai_2026_renewal_changes_law_critical_honeycomb_walks
• openai_2026_renewal_changes_law_critical_honeycomb_walks / theorem_3_3
• openai_2026_renewal_changes_law_critical_honeycomb_walks / theorem_6_4
• openai_2026_renewal_changes_law_critical_honeycomb_walks / theorem_8_1
• openai_2026_renewal_changes_law_critical_honeycomb_walks / theorem_8_2