Scaling limit of random recursive square subdivision

Definitions and mathematical statement

Start with $\mathcal Q_0=\{[0,1]^2\}$. At each step choose one square uniformly from the current collection, remove it and replace it by its four congruent quarter-squares. After $n$ subdivisions there are $1+3n$ current squares. This defines all stages on the same recursive probability space.

For a precise literal connected-union reading, let \[ D_n=\min\{|\mathcal A|:\mathcal A\subseteq\mathcal Q_n, \ \bigcup_{Q\in\mathcal A}Q\text{ is connected and contains }(0,0),(1,1)\}. \] The current squares here are closed, so corner contacts count as connectivity. If the source instead requires adjacency along a positive-length edge or another crossing convention, it defines a different quantity and must be specified.

The first question asks for a deterministic positive sequence $(a_n)$ such that $D_n/a_n$ converges to a nondegenerate random variable. In the usual distributional interpretation the target is \[ \exists a_n>0\text{ deterministic}\ \exists D\text{ finite and not almost surely constant}:\quad D_n/a_n\ \Longrightarrow\ D. \] The excerpt does not specify the convergence mode; almost-sure or in-probability convergence would be stronger requests and are not asserted.

The second question asks whether a shortest crossing stabilizes as subdivision continues. A shortest square collection need not be unique, and the collections themselves change when their squares are subdivided. A precise stabilization notion must specify tie selection and whether it concerns a geometric path, coarse ancestral choices, or another projection of the changing crossings. No eventual equality of current-square lists or canonical path is silently imposed.

The supplied source-side and bounded modern audits report no complete proof or counterexample. The summary calls the notes "2024", while the supplied preserved copy has a 2020 archive timestamp; no publication date is inferred from that discrepancy. The open label remains a qualified source assessment rather than independently verified current status.

Short English statement

Randomly subdivide one current square into four at each step. Does the minimum square-count crossing between opposite corners have a deterministic nondegenerate scaling limit, and do shortest crossings stabilize in an appropriate sense? Connectivity, convergence and stabilization conventions need clarification.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10000013$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Itai Benjamini, Coarse Geometry and Randomness, Saint-Flour notes, Open problem 1.73, PDF page 16: https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page=16. Preserved copy supplied at https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf. Supplied text extraction checked against rendered PDF, accessed 2026-07-29.
• The provenance reports cross-document deduplication, source-side review and a bounded modern audit with no complete resolution located; no separate resolution publication is supplied.