Extend maximal-persistence results to Vietoris–Rips complexes

Establish analogous universal asymptotic results for the maximal multiplicative persistence of homological cycles in Vietoris–Rips filtrations, including an appropriate proof of the upper bound and the necessary geometric statements.

Background

The paper analyzes extreme multiplicative persistence for Čech complexes and notes that its upper-bound argument relies on geometric properties specific to Čech filtrations. The authors expect a similar theory for Vietoris–Rips complexes but explain that the current proof does not extend because the relevant geometric statements are unavailable in that setting.

The problem is explicitly unresolved: it concerns both proving analogous results for Vietoris–Rips complexes and developing a different method for the upper bound. Its scope is concrete rather than a general future-work aspiration.

References

While we conjecture that similar results apply to the Vietoris-Rips complex, the proof for the upper bound, and in particular the geometric statements in Section \ref{sec:geom} do not apply. Therefore, a different approach is required which will remain as future work.

— A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes  (2609.19474 - Bobrowski et al., 16 Sep 2026) in Section 2, subsection “Filtrations of geometric complexes”

While we conjecture that Theorem \ref{thm:main} can be generalized far beyond $D$, it seems that the proofs will have to involve intricate technical details on a case-by-case basis.

— A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes  (2609.19474 - Bobrowski et al., 16 Sep 2026) in Remark following Theorem 3.1 (Section 3, “Main Results”)