Macroscopic Pivotals Without Infinite-Cluster Exceptional Times

Definitions and mathematical statement

For notation only, let $G=(V,E)$ denote the underlying infinite graph, which is not specified in the supplied record. An edge configuration is an element $\omega\in\{0,1\}^{E}$; an edge $e$ is open when $\omega(e)=1$. An open cluster is a connected component of the graph formed by the open edges. Define \[ \mathcal I_G=\{\omega:\text{an open cluster contains infinitely many vertices}\}. \] Critical FK heat-bath dynamics refers to dynamics for the Fortuin--Kasteleyn random-cluster model, with cluster-weight parameter $q$ and edge parameter at criticality, in which an updated edge is resampled according to its conditional random-cluster distribution given the other edges. The graph, infinite-volume measure, boundary conditions, update rates, initial law, and time domain are not specified in the record.

For an event $A\subseteq\{0,1\}^{E}$, an edge is pivotal at $\omega$ if changing only its state changes whether $A$ occurs. The stronger phrase "macroscopic pivotals exist" requires an event, a spatial scale or scaling limit, and an existence criterion; these are not supplied.

Source question. Does there exist a regime \[ q\in(q^*,4), \qquad\text{described equivalently in the source as}\qquad \kappa\in(4,\kappa^*) \] for critical FK heat-bath dynamics in which macroscopic pivotals exist but there is no exceptional time at which an infinite open cluster exists? Here $\kappa$ is the parameter in the source's $\mathrm{SLE}_{\kappa}$ description.

If $(\omega_t)_{t\in T}$ denotes the intended process, the absence of any infinite-cluster time would be expressed as \[ \mathbb P\!\left(\exists t\in T:\omega_t\in\mathcal I_G\right)=0. \] This is a conditional rendering of the intended absence assertion, not a fully specified conjecture: the law $\mathbb P$ and time domain $T$ remain unspecified. The usual exceptional-time interpretation additionally presupposes that an infinite cluster has probability zero at each fixed deterministic time; the supplied record does not separately state this hypothesis. The thresholds $q^*$ and $\kappa^*$, their relationship, and the correspondence between $q$ and $\kappa$ are also unspecified. Consequently the asserted equivalence of parameter intervals and the precise quantifiers defining the regime cannot be resolved from this record.

Short English statement

The question asks whether critical FK heat-bath dynamics can have macroscopic pivotal edges while almost surely never producing an infinite cluster, for a regime described as $q\in(q^*,4)$, or equivalently $\kappa\in(4,\kappa^*)$. The thresholds, model specifications, and meaning of macroscopic pivotality require clarification before this becomes a complete mathematical statement.

Sources

• Local registry: supplied UnsolvedMath record, permanent ID $30001803$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Repository registry locations are data/unsolvedmath/problems.json and catalog/problems.json; their contents were not independently inspected for this draft.
• Supplied source citation: Stochastic Analysis (2011), Oberwolfach Reports, DOI 10.4171/owr/2011/29, pp. 1573--1635. Original Oberwolfach report: https://doi.org/10.4171/owr/2011/29. The record attributes the question to the workshop report and supplies no individual proposer.
• Supplied literature source labeled "Steif--dynamical percolation": https://annals.math.princeton.edu/2010/171-2/p01. No further bibliographic attribution is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says the report presents this exceptional-time phenomenon for critical FK models as a conjecture, and says the cited dynamical-percolation results do not establish the stated parameter regime. This is supplied dated curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. These are supplied verification claims, not verification performed for this draft.