Fluctuations of Poisson-Line Cell Boundary Paths

Definitions and mathematical statement

Put $a_n=(-n/2,0)$, $b_n=(n/2,0)$, and let $[a_n,b_n]$ be their Euclidean joining segment. The source refers to a Poisson-line-process cell $C_n$ containing both endpoints and to clockwise and counterclockwise paths on $\partial C_n$ between them. For the intended construction, it asks, as $n\to\infty$, about the fluctuations of the boundary-path length, its maximum lateral deviation from the direct segment, and the discrepancy between these two candidate paths and the true network geodesic.

Once a path $\Gamma_n$ is actually specified, relevant observables include its Euclidean arclength $L_n$ and the maximum distance \[ D_n=\sup_{z\in\Gamma_n}\operatorname{dist}(z,[a_n,b_n]). \] If lateral deviation instead means perpendicular distance from the supporting horizontal line, the corresponding quantity is $\sup_{(x,y)\in\Gamma_n}|y|$, which need not be the same if the path extends past the endpoints. The source does not fix this convention, a centering and scaling for $L_n$, or the metric used to compare entire paths with the geodesic.

There is a construction gap in the literal wording. Two fixed points need not lie in one cell of an ordinary Poisson line tessellation; moreover, in the usual diffuse line-process setting they lie in cell interiors rather than on cell boundaries, so a boundary arc cannot simply start and end at those points. A conditioning, deletion of separating lines, endpoint-attachment rule, or other construction may be intended, but none is supplied or selected here. The network and the meaning of a geodesic from these endpoints likewise require an attachment and length convention. Consequently the requested fluctuation laws remain dependent on recovering the exact model, not on a silently modified Poisson tessellation.

Short English statement

Describe boundary-route length fluctuations, maximum deviation, and the difference from shortest network routes in the intended Poisson-line model. The extraction omits how one cell and boundary paths are attached to the two fixed endpoints.

Sources

• Local UnsolvedMath registry, permanent ID $30001079$, source code catalogue entry, "Fluctuations of Poisson-Line Cell Boundary Paths." No individual proposer is supplied.
• New Perspectives in Stochastic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/47, pp. 2655--2702, listed as "OWR Report 47": https://doi.org/10.4171/owr/2008/47. This is the sole supplied external reference.
• Supplied literature triage, checked 2026-08-21: status "open." It reports known asymptotic mean geometry in the cited model but calls the requested random-variation and vertical-deviation theory unresolved in the extracted general form. The mean formula and model details are not reproduced. This is supplied dated curation, not independently verified; time-sensitive status and cited claims require independent verification.