Persistence Times of Dynamical-Percolation Crossings

Definitions and mathematical statement

The supplied question is: "How long may it take for a crossing event to disappear in dynamical percolation?" The record does not specify enough model data to determine a unique persistence-time problem. The following gives a conditional formulation, with its unspecified parameters displayed explicitly.

Fix a graph $G$, a subgraph $D$, and two designated boundary sets $B_1,B_2\subseteq V(D)$. Specify either site or bond percolation. Let $(\omega_t)_{t\geq 0}$ be the chosen dynamical-percolation process, with its percolation parameter, update mechanism, update rates, and boundary conditions specified. Define the crossing event \[ A=\{\omega:\text{there is an open path in }D\text{ joining }B_1\text{ to }B_2\}. \] Here openness is interpreted according to the selected site or bond model. For a specified initial distribution $\mu$ satisfying $\mu(A)=1$, define the first disappearance time and its tail by \[ T_A=\inf\{t\geq 0:\omega_t\notin A\},\qquad S_{A,\mu}(t)=\mathbb P_\mu(T_A>t),\quad t\geq 0, \] with $\inf\varnothing=+\infty$. Conditioning a stationary initial distribution on $A$ is one possible initial convention, but the source does not specify that convention.

Conditional mathematical target. For the dynamical-percolation model and crossing geometry intended by the source, determine the distribution of $T_A$, or quantitative bounds and asymptotic behavior for $S_{A,\mu}(t)$. If a family of crossing regions is intended, its indexing, time normalization, and order of limits must also be specified. This is a formulation template, not a claim that one bound should hold for every graph, parameter, geometry, update rule, or initial law. The source does not identify whether the requested result is an expectation, a tail estimate, a sharp asymptotic law, or a statement uniform over scales.

The supplied background discusses critical dynamical percolation and box crossings, but the question itself does not impose criticality or identify a lattice, dimension, or box. Its dated literature assessment reports superpolynomial decay for box-crossing persistence and exponential tails for the first exceptional infinite-cluster time. The latter is a different stopping-time question and is not identified here with $T_A$. The record labels the problem partially solved; this label and the reported bounds are supplied curation, not independently verified mathematical conclusions.

Short English statement

Starting with an open crossing, ask how long it remains present under dynamical percolation. A precise persistence-tail question requires the graph, crossing geometry, percolation parameter, dynamics, initial law, and time scale. These are not fixed by the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $30002184$, source problem code catalogue entry, title "Persistence Times of Dynamical-Percolation Crossings." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2012$; no individual proposer is identified.
• Supplied report citation: Scaling Limits in Models of Statistical Mechanics (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/44, pp. 2657--2745. https://doi.org/10.4171/owr/2012/44. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied citation uses 2013, while the DOI and proposed-year metadata refer to 2012; these details are retained without reconciliation.
• The record labels its statement exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification assertion; no independent source inspection is claimed here. No particular problem page within the cited report is supplied.
• Supplied literature reference: Hammond--Mossel--Pete, "exit-time tails." https://doi.org/10.1214/EJP.v17-2229.
• Supplied literature reference: Hammond--Pete--Schramm, "exceptional-set local time." https://arxiv.org/abs/1208.3826.
• Literature assessment checked at 2026-08-21: the supplied status is partially_solved. The assessment reports superpolynomial decay for persistence of a box crossing in critical dynamical percolation and exponential tails for the first exceptional infinite-cluster time, while stating that these results do not determine a sharp general tail law. This is supplied dated curation derived from the OWR collection, not independently verified. The record expressly requests independent verification of time-sensitive status and cited claims.