Finiteness and Tail Decay of Infinite-Noodle Components

Definitions and mathematical statement

Let $C(0)$ denote the connected component containing the distinguished origin $0$ in the random structure called the infinite-noodle model in the supplied record, and let $\mathbb P$ denote its probability law. Write $X=|C(0)|$ for its size under the intended model's size convention, allowing $X=\infty$. The supplied record does not define the model, its probability law, or what is counted by component size. Consequently, the following formulation retains those unresolved source-dependent conventions and is not a self-contained specification of the random object.

The two conjectural assertions to be established are \[ \mathbb P\left(C(0)\text{ is finite}\right)=1 \] and, with $\alpha=(2\sqrt{2}-1)/7$, \[ \mathbb P(X\geq k)=k^{-\alpha+o(1)} \qquad\text{as }k\to\infty,\quad k\in{\mathbb{Z}}_{\geq1}. \] Precisely, the second assertion means that there exists a real-valued function $r$ on the positive integers, with $r(k)\to0$, such that \[ \mathbb P(X\geq k)=k^{-\alpha+r(k)} \] for every sufficiently large positive integer $k$. No assertion is made here about a multiplicative asymptotic constant.

The supplied original excerpt attributes almost-sure finiteness to its reference [4] and the tail conjecture to its reference [1], Conjecture 1.10. The corresponding numbered bibliography is absent, so these reference labels cannot be reliably assigned to the supplied literature links.

Short English statement

In the intended infinite-noodle model, is the component containing the origin finite almost surely, and does its size have tail probability $k^{-(2\sqrt{2}-1)/7+o(1)}$ as positive integers $k$ tend to infinity? The model and the convention for measuring component size require clarification from the cited source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006519$, title: Finiteness and Tail Decay of Infinite-Noodle Components; exact source problem code: catalogue entry. The record labels the problem open and supplies no named proposer.
• Enumerative Combinatorics (2026), Oberwolfach Reports, pp. 81--156, DOI 10.4171/owr/2026/2. Supplied report provenance: https://doi.org/10.4171/owr/2026/2. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Féray--Thévenin, Components in meandric systems and the infinite noodle. Supplied literature reference: https://arxiv.org/abs/2201.11572.
• Curien et al., Uniqueness of the infinite noodle. Supplied literature reference: https://arxiv.org/abs/1701.01083.
• Supplied literature assessment dated 2026-08-22: the component of zero is not known to be finite almost surely, and the stated power-law tail conjecture remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The source itself requests independent verification of time-sensitive status and cited claims.
• The registry assigns the statement the label corrected_verified, explaining that only formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. This supplied editorial label does not establish independent mathematical verification.