The diameter of a long‐range percolation graph

We consider the following long‐range percolation model: an undirected graph with the node set {0, 1, … , N}d, has edges (x, y) selected with probability ≈ β/∥x − y∥s if ∥x − y∥ > 1, and with probability 1 if ∥x − y∥ = 1, for some parameters β, s > 0. This model was introduced by Benjamini and Berger [2], who obtained bounds on the diameter of this graph for the one‐dimensional case d = 1 and for various values of s, but left cases s = 1, 2 open. We show that, with high probability, the diameter of this graph is Θ(log N/log log N) when s = d, and, for some constants 0 < η1 < η2 < 1, it is at most N when s = 2d, and is at least N when d = 1, s = 2, β < 1 or when s > 2d. We also provide a simple proof that the diameter is at most logO(1) N with high probability, when d < s < 2d, established previously in [2].