Sharp Tail Decay for Persistent Critical Crossings

Definitions and mathematical statement

Let $Q$ be a square with designated left and right sides. For the percolation model intended by the source, define \[ C_Q=\{\omega:\text{there is an open crossing of }Q\text{ joining its left and right sides}\}. \] Let $(\omega_s)_{s\geq0}$ denote the intended critical dynamical percolation process, and write \[ p_Q(t)=\mathbb P\!\left[\omega_s\in C_Q\text{ for every }s\in[0,t]\right],\qquad t\geq0. \] The probability displayed in the source is unconditional; it does not explicitly condition on $\omega_0\in C_Q$.

The supplied conjectural question is whether \[ p_Q(t)\leq\exp\!\left(-t^{2/3+o(1)}\right)\qquad(t\to\infty). \] Interpreting this as an upper bound with an unspecified vanishing exponent correction, its quantified meaning is \[ \forall\varepsilon>0\ \exists T_\varepsilon>0\ \forall t\geq T_\varepsilon: \qquad p_Q(t)\leq\exp\!\left(-t^{2/3-\varepsilon}\right). \] This interpretation supplies no matching lower bound.

The formulation remains provisional: the supplied record does not specify the lattice or continuum model, the critical parameter, the initial law, the update mechanism and time normalization, boundary conditions, or whether a scaling limit is intended. It also does not state whether the bound is for a fixed square and fixed model or is uniform over spatial scales or discretizations. These choices are required to identify the probability law and the intended scope of the conjecture; none is imposed here.

Short English statement

The question asks whether the probability that a critical dynamical percolation configuration has a left--right crossing of a square at every time from $0$ to $t$ has the conjectured upper bound $\exp(-t^{2/3+o(1)})$. The underlying model, probability law, time normalization, and scale quantifiers require clarification.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002185$, title "Sharp Tail Decay for Persistent Critical Crossings," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2012 and names no proposer.
• Supplied report citation: Scaling Limits in Models of Statistical Mechanics (2013), Oberwolfach Reports, pp. 2657--2745, DOI 10.4171/owr/2012/44. Supplied source URL and literature reference "Oberwolfach Report 44, dynamical percolation": https://doi.org/10.4171/owr/2012/44.
• Supplied literature reference: Hammond--Mossel--Pete, "exit-time tails," https://doi.org/10.1214/EJP.v17-2229.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the displayed decay as conjectural possible sharp decay. It states that the cited dynamical-percolation results establish exponential or superpolynomial bounds in related regimes, rather than this asymptotic. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified 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 verification description is supplied metadata and has not been independently checked for this draft.