Contact Processes on Exponential-Growth Dynamic Graphs

Definitions and mathematical statement

The supplied record concerns contact processes on dynamical percolation on graphs of exponential growth. It does not specify the graph class, the stochastic dynamics, the parameters, the initial conditions, or the definition of exponential growth being used.

For notation only, if an infection process has been specified, write $\eta_t\subseteq V(G)$ for its infected vertices at time $t\geq 0$. Survival means the event \[ \{\eta_t\neq\varnothing\text{ for every }t\geq 0\}. \] A positive survival probability must be evaluated under a specified law, including initial infection and percolation configurations; these are absent from the extract. The parameter or parameters defining criticality are likewise unspecified.

The explicit conjectural direction in the supplied extract is that the authors' results extend to a certain untreated parameter regime on exponential-growth graphs. Schematically, let $\mathcal G$ denote the intended graph class, let $U(G)$ denote that untreated regime, and let $\mathcal R(G,\theta)$ denote the conclusions of the results being extended. The requested extension has the form \[ \forall G\in\mathcal G\;\forall\theta\in U(G),\qquad \mathcal R(G,\theta). \] Here $\mathcal G$, $U(G)$, the parameter tuple $\theta$, and $\mathcal R$ are placeholders for information missing from the supplied record, not definitions of a complete mathematical conjecture.

The registry summarizes the goal as establishing survival and criticality throughout the untreated regime. The extract itself does not identify the relevant survival or criticality conclusions. Its further statement that the work mainly focuses on the supercritical parameter regime does not specify a conjecture outside that regime. Consequently, a precise, self-contained theorem target cannot be recovered from the supplied evidence.

Short English statement

Extend the authors' results for contact processes on dynamical percolation to an untreated parameter regime on exponential-growth graphs. The graph assumptions, process law, excluded regime, and conclusions to extend are missing. The reference to work beyond the supercritical regime gives no precise additional target. This is an incomplete statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005046$, source problem code catalogue entry, title "Contact Processes on Exponential-Growth Dynamic Graphs," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2022.
• Supplied report citation: Population Dynamics and Statistical Physics in Synergy (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/12, pp. 577--656. Supplied source and literature link: https://doi.org/10.4171/owr/2022/12. The extract includes a page marker for p. 645. The report contents were not independently consulted for this draft.
• Supplied literature assessment dated 2026-08-22: the extension to the untreated parameter regime and further work beyond the supercritical regime are described as unresolved, and the registry status is "open." This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft.