On the intersection graph of random caps on a sphere

Drop N spherical caps, each of area 4π·p(N), at random on the surface of a unit sphere, and let Gp denote the intersection graphs of these random caps. Among others, we prove the following: (1) If N(Np)n−1→0 as N→∞, then Pr(Gp has no component of order ≥n)→1, while if N(Np)n−1→∞ then Pr(Gp has an n-clique)→1 as N→∞. (2) If p<1−ε4NlogN, ε>0 then Pr(δ=0)→1, while if p>1+ε4NlogN then for any positive integer n, Pr(δ≥n)→1 as N→∞, where δ denotes the minimum degree of Gp. (3) If p=14N(logN+x) then the number of isolated vertices of Gp is asymptotically (N→∞) distributed according to Poisson distribution with mean e−x. (4) If p>1+ε2NlogN, then Pr(Gp is 2-connected)→1 as N→∞.