Abstract
DropN random caps all of the same angular radius\(\theta = c\sqrt {\tfrac{1}{N}\log N} \) on a unit sphere. LetU denote the part of the surface covered by these caps. We prove that if\(c > \sqrt 2 \), then the probability thatU is connected tends to 1 asN→∞, while ifc<1, then the probability thatU is connected tends to 0 asN→∞.
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Maehara, H. When does the union of random spherical caps become connected?. Ann Inst Stat Math 56, 397–402 (2004). https://doi.org/10.1007/BF02530553
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DOI: https://doi.org/10.1007/BF02530553