When does the union of random spherical caps become connected? (Q1768136)

Let \(U\) denote the part of the surface covered by \(N\) random spherical caps \(c_1, c_2,\dots,c_N\) of the same angular radius \(\Theta\) placed on the surface of a unit sphere in 3-space. It is proved that if \(c>\sqrt 2\), then the probability that \(U\) is connected tends to 1 as \(N\to 0\), while if \(c<1\), then the probability that \(U\) is connected tends to 0 as \(N\to\infty\).