Percolation Structure in Nearly Optimal Disk Packings

Definitions and mathematical statement

A packing of unit disks in the Euclidean plane is a family of closed disks $\{\overline B(x,1):x\in X\}$, where $X\subset{\mathbb{R}}^2$ and $\lVert x-y\rVert\geq 2$ for distinct $x,y\in X$; thus disk interiors are disjoint. For a parameter $\varepsilon>0$, define the undirected graph \[ G_\varepsilon(X)=(X,E_\varepsilon),\qquad E_\varepsilon=\left\{\{x,y\}:x,y\in X,\ x\ne y,\ \lVert x-y\rVert\leq 2+\varepsilon\right\}. \] An infinite connected component means a connected component containing infinitely many centers. The triangular-lattice packing of unit disks has covered-area density $\rho_\triangle=\pi/\sqrt{12}$.

The supplied question asks for quantitative density assumptions, expressing proximity to $\rho_\triangle$, that force $G_\varepsilon(X)$ to have an infinite connected component. A threshold formulation to be clarified is the following: for a specified packing class $\mathcal P$, a specified covered-area density functional $D$, and a specified $\varepsilon>0$, does there exist $\delta>0$ such that \[ \forall X\in\mathcal P,\qquad D(X)>\rho_\triangle-\delta \ \Longrightarrow\ G_\varepsilon(X)\text{ has an infinite connected component}? \] If such a formulation is intended, determine sufficient quantitative bounds on $\delta$, including their dependence on $\varepsilon$ and any model parameters.

This is an unresolved formulation schema, not an assertion that such a threshold exists. The record does not specify the density functional, the admissible packing class, or the quantification over $\varepsilon$. Its statements concern packings without an explicit probability model, whereas its literature assessment identifies a conjecture about random packings and refers to Gibbs states. If a probabilistic formulation is intended, the probability law and the required probability of percolation must also be specified. No deterministic or almost-sure conclusion is silently substituted for the other.

Short English statement

Join distinct centers of nonoverlapping unit disks whenever their distance is at most $2+\varepsilon$. The question asks how close the packing density must be to the triangular-lattice value to force an infinite connected component. A precise threshold question requires a density definition, a class of packings or a probability model, and quantifiers for $\varepsilon$; these are missing from the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $30004119$, title: Percolation Structure in Nearly Optimal Disk Packings; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2019. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Mathematics of Crystallisation (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/18, pp. 1071--1111. https://doi.org/10.4171/owr/2019/18. The literature-source label for this same link is "Oberwolfach Report 18/2019, Percolation of two-dimensional hard disks." The supplied 2020 workshop label and 2019 report designation are retained without resolving their discrepancy.
• Supplied literature source: Bowen--Lyons--Radin--Winkler, A solidification phenomenon in random packings. https://doi.org/10.1137/050647785. The registry attributes the conjectural identification to these authors in its literature assessment; it does not identify an individual proposer of the registry problem.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified source consultation or a current literature determination. The supplied assessment labels the problem open, identifies it with the Bowen--Lyons--Radin--Winkler percolation conjecture, and says that available density and Gibbs-state results support it but do not prove the assertion for every near-optimal packing model. The background explicitly requests independent verification of time-sensitive status and cited claims.