Integrated Super-Brownian Scaling of High-Dimensional Incipient Clusters

Definitions and mathematical statement

Let $\mathcal C_{\mathrm{IIC}}$ denote the random cluster referred to in the supplied record as the high-dimensional incipient infinite cluster (IIC). The record does not specify the underlying percolation model, ambient space, dimension range, or construction of its law. Let $\mathrm{ISE}$ denote integrated super-Brownian excursion, the proposed limiting random measure; its precise construction and normalization are not supplied.

Source question and conjecture. Identify the scaling limit of the entire high-dimensional incipient infinite cluster. The supplied statement conjectures that a properly rescaled cluster converges to $\mathrm{ISE}$.

Unresolved formulation. This is a source-level conjecture, not a fully specified convergence assertion. The record gives no indexed family of random objects, scaling parameter, spatial or mass normalization, representation of the cluster, state space, topology, or mode of convergence. It also does not explain how the entire infinite cluster is to be represented and normalized to yield the proposed ISE limit, or whether any truncation or conditioning is intended. No such operation is assumed here. Consequently, the hypotheses and quantifiers required for a precise mathematical convergence statement cannot be recovered from the supplied record.

The supplied dated literature assessment describes the extract as ending before the object to be identified is retained, whereas the supplied statement fields explicitly name both the entire IIC and ISE. This discrepancy remains unresolved.

Short English statement

The supplied conjecture asks whether a suitably rescaled entire high-dimensional incipient infinite cluster has integrated super-Brownian excursion as its scaling limit. The model, dimension assumptions, rescaling, and meaning of convergence are missing, so the record does not determine a precise convergence statement. Its supplied open label is not independently verified.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30002186$, source problem code catalogue entry, title Integrated Super-Brownian Scaling of High-Dimensional Incipient Clusters; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2012.
• Supplied source 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 supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise page or passage for this problem is supplied.
• Supplied literature source: "Oberwolfach Report 44, source extract," https://doi.org/10.4171/owr/2012/44. Literature checked at: 2026-08-21. This is supplied dated curation, not independently verified. It labels the problem open and assesses the extract as incomplete, unresolved, and unclassifiable as solved under the queue rule. The background requests independent verification of time-sensitive status and cited claims.
• The registry labels the 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. These are supplied verification assertions; this draft does not independently verify them or reconcile them with the dated assessment of an incomplete extract.