Determine the self-avoiding-walk connective constant $C_k$ in $\mathbb{Z}^k$ (Erdős #528)
Statement
Let $f(n,k)$ count the self-avoiding walks of $n$ steps starting at the origin in the integer lattice $\mathbb{Z}^k$ (a self-avoiding walk is a lattice path that visits no vertex more than once). Determine the connective constant $$C_k=\lim_{n\to\infty} f(n,k)^{1/n}.$$
Acceptance. FULLY RESOLVES: an exact closed-form determination of $C_k$ for some $k\ge2$ together with a proof (for instance establishing a conjectured algebraic value), OR a proof of the exact asymptotic expansion of $C_k$ as $k\to\infty$ to all orders. Provide a complete written proof, machine-checkable (Lean/Coq) preferred. ADVANCES (steer away from duplicating the existing SciNet square-lattice and cubic-lattice bound problems): rigorously narrow the two-sided enclosure of $C_k$ for a dimension $k\ge4$ not covered by those problems, with a reproducible enumeration/transfer-matrix certificate strictly better than the bounds stated in the background; OR sharpen a coefficient of Kesten's expansion $C_k=2k-1-\tfrac{1}{2k}+O(1/k^2)$ with proof; OR exactly enumerate the series $f(n,k)$ to a new record length $n$ for some fixed $k$ (deliver the code and counts). Deliver the proof/closed form, or the enumeration code plus the attained series/bounds and a certificate of correctness.
Background
Posed by Erdős [Er61, p.254]; listed as open on erdosproblems.com/528 (fetched 2026-07-13, status 'open', tagged 'geometry'). The limit $C_k$, known as the connective constant, was shown to exist by Hammersley and Morton [HM54]; the elementary bounds $k\le C_k\le 2k-1$ are immediate. Kesten [Ke63] proved the high-dimensional asymptotic $C_k=2k-1-\tfrac{1}{2k}+O(1/k^2)$, refined further by Clisby, Liang, and Slade [CLS07]. In the plane ($k=2$, the square lattice) Conway and Guttmann [CG93] showed $C_2\ge 2.62$ and Alm [Al93] showed $C_2\le 2.696$; Jacobsen, Scullard, and Guttmann [JSG16] computed the first digits, $C_2=2.6381585303279\cdots$ (OEIS A387897), and the algebraic value defined by the root of $13x^4-7x^2-581=0$ (OEIS A156816) is a candidate closed form in this circle of questions. No exact value of $C_k$ is known for any $k$, and the exact $k\to\infty$ asymptotic expansion beyond Kesten's leading terms is open. Note: two existing SciNet problems already ask to narrow the rigorous two-sided bounds on the connective constant for the specific square lattice ($\mathbb{Z}^2$) and simple cubic lattice ($\mathbb{Z}^3$); the present problem is the general-dimension version, most distinctive for dimensions $k\ge4$ and for the exact asymptotics in $k$. Attacker's tool: exact enumeration of self-avoiding walks via finite-lattice / transfer-matrix methods to extend the series $f(n,k)$ and tighten rigorous enclosures of $C_k$, together with asymptotic-series analysis of Kesten's expansion.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #528 (T. F. Bloom) | website |
| REF-02 | OEIS A387897 — Decimal expansion of the connective constant of the square lattice | website |
| REF-03 | OEIS A156816 — Decimal expansion of the positive root of 13x^4 - 7x^2 - 581 = 0 | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.