On random points in the unit disk

Let n be a positive integer and λ > 0 a real number. Let Vn be a set of n points in the unit disk selected uniformly and independently at random. Define G(λ, n) to be the graph with vertex set Vn, in which two vertices are adjacent if and only if their Euclidean distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ lnn/n and let X be the number of isolated points in G(λ, n). We prove that almost always X ∼ n1−c2 when 0 ≤ c < 1. It is known that if λ = √ (lnn+ φ(n))/n where φ(n) → ∞, then G(λ, n) is connected. By extending a method of Penrose, we show that under the same condition on λ, there exists a constant K such that the diameter of G(λ, n) is bounded above by K · 2/λ. Furthermore, with a new geometric construction, we show that when λ = c √ lnn/n and c > 2.26164 · · · , the diameter of G(λ, n) is bounded by (4 + o(1))/λ; and we modify this construction to yield a function c(δ) > 0 such that the diameter is at most 2(1 + δ + o(1))/λ when c > c(δ). ∗This is a preprint of an article accepted for publication in Random Structures and Algorithms c ©2005. †Partially supported by NSF grant DMS-9977354. ‡Partially supported by NSF grant DMS-0245526, DMS-0308827 and a Sloan Fellowship. The author is also affiliated with Dalian University of Technology.