Record asymptotics for uniform two-dimensional coverage thresholds

Determine the asymptotic record-count and record-time behavior of the uniform coverage threshold sequence for a compact region equal to the sampling region in dimension two: establish whether, for k-coverage with k\geq 2, the asymptotics in equations (0720a), (0720b), and (0720c) hold, and characterize the corresponding limiting record-count constant for k=1 as a perimeter-dependent weighted average of 1 and 1/2.

Background

The paper studies lower records of coverage thresholds as independent random points are added to a bounded region. For a coverage threshold R_{n,k}, a record occurs when the threshold decreases after the arrival of a new point. The authors establish almost-sure asymptotics for regions strictly inside the sampling region and for the case B=A in dimensions d=1 and d\geq 3, with different constants determined by whether the extremal uncovered location lies in the interior or near the boundary.

The uniform case B=A in dimension d=2 is unresolved because the largest uncovered hole may be either an interior hole or a boundary hole. The competition between these two types of extremal locations makes the limiting record frequency more delicate. The authors conjecture the established boundary-dominated asymptotics for k\geq 2, while predicting a perimeter-dependent mixture of interior and boundary contributions for k=1.

References

When d=2 (in the uniform case with B=A) we do not present any results, but based on Theorem 3.2 we conjecture that if k \geq 2 then 0720a, 0720b and 0720c hold, while if k=1 then the limiting value of 2 N_{n,k}/(\log n)2 will be a weighted average of 1 and 1/2, with the weights depending on the perimeter of A.

— Record times for coverage thresholds and maximal spacings  (2608.19104 - Penrose, 19 Aug 2026) in Section 1, immediately following Theorem 3 in Section 2, paragraph beginning “In the uniform case with B=A”