Long-tailed degree distribution of a random geometric graph constructed by the Boolean model with spherical grains

We consider a random geometric graph constructed by the homogeneous Boolean model with spherical grains in R, d ≥ 2; that is, a node of the graph corresponds to a germ of the Boolean model and there is an edge between two nodes when their grains intersect with each other. We show that, when the radius distribution of grains is long-tailed, so is the degree distribution of the graph. Our result includes as special cases that, if the radius distribution is regularly varying with index −α with α > d, then the degree distribution is regularly varying with index −α/d and, in the case of d = 2, if the radius distribution is long-tailed with the second moment, then the degree distribution is square-root insensitive. In the proof, a subclass of long-tailed distributions — called x1−p-insensitive distributions with p ∈ (0, 1) — plays a key role.