Asymptotic constants for higher homology degrees

Determine whether there exist constants c_{s,p,d} such that the expected total degree-p homology of the growing s-lunes generated by n uniformly sampled points in [0,1]^d with random (s+1)-coloring is asymptotic to c_{s,p,d} n^{1-1/d} as n tends to infinity.

Background

The paper establishes an asymptotic constant only for the degree-zero persistence information associated with growing s-lunes in the planar setting. Through the connection between lunar EMSTs and chromatic persistence, the authors identify higher homology degrees as a natural extension.

They specifically ask whether the same asymptotic scaling law persists for degree-p homology in arbitrary dimension d, with constants depending on the number of colors, homology degree, and ambient dimension. The case s=0, p=1, d=2 is cited as already resolved in earlier work.

References

Do asymptotic constants exist for homology degrees beyond $0$? Specifically, are there constants $c_{s,p,d}$ such that the expected total degree-$p$ homology of growing $s$-lunes of $n$ points sampled uniformly at random in $[0,1]d$ and randomly $(s+1)$-colored is $c_{s,p,d} \cdot n{1 - \sfrac{1}{d}$ in the limit, when $n$ goes to infinity?

— Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs  (2608.27118 - Draganov et al., 27 Aug 2026) in Section 5, Discussion