Bobrowski–Skraba strong universality conjecture

For every admissible random point-cloud sampling model, ambient dimension $d$ , homological degree $k$ , and filtration type, let $(b_{n,i},d_{n,i})_{i=1}^{N_n}$ be the persistence pairs with $0<b_{n,i}<d_{n,i}$ , and define the empirical measure of log-log persistence ratios by $\mu_n=\frac{1}{N_n}\sum_{i=1}^{N_n}\delta_{\log\!\log(d_{n,i}/b_{n,i})}$ . The strong universality conjecture asserts that, after the prescribed data-dependent additive centering $c_n$ , the centered measures $\frac{1}{N_n}\sum_{i=1}^{N_n}\delta_{\log\!\log(d_{n,i}/b_{n,i})-c_n}$ converge weakly to one and the same universal limiting probability law, independently of the sampling model, $d$ , $k$ , and the filtration; the conjectured universal law is a left-skewed Gumbel distribution.

References

Primary source

arXiv

Additional references

Near-diagonal asymptotics and the failure of strong universality in random Čech persistence — arXiv — Eunwoo Heo

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

An unrefereed preprint claims to refute the conjecture by showing that the proposed universal law differs between the square and cube, but the claim has not been independently verified.

Bobrowski and Skraba proposed a strong universality principle for persistence-ratio distributions in random point-cloud filtrations. It predicts universality beyond the fixed dimension, homological degree, and filtration covered by the weaker conjecture.

Known results

The 2022 formulation presented strong universality as a conjecture supported by experiments.
A 2024 paper claims the weaker result: for fixed dimension $d$ , degree $k$ , and filtration type, iid point processes have a universal limiting empirical distribution.

October 2026 counterexample

On October 6, 2026, a report summarized Eunwoo Heo's preprint as showing that degree-one Čech persistence has limiting quantities $\pi^2/32$ for the square and $4/\pi^2-1/4$ for the cube, forcing different limiting laws and hence contradicting strong universality. A separate 2026 preprint claims a broader universality theorem but does not address these quantities.

Current status (as of October 2026): the weaker fixed-parameter universality result is claimed in the literature, while strong universality is claimed refuted by Heo's counterexample, whose proof remains unverified.

Sources

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