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 approximate to beta/parallel to\x - yparallel to(s) if parallel tox - yparallel to > 1, and with probability 1 if parallel tox - yparallel to = 1, for some parameters beta, 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 Theta(log N/log log N) when s = d, and, for some constants 0 < η(1), < eta(2) < 1, it is at most N-η2 when s = 2d, and is at least N-η1 when d = 1, s = 2, β < 1 or when s > 2d. We also provide a simple proof that the diameter is at most log(O(1)) N with high probability, when d < s < 2d, established previously in [2]. (C) 2002 Wiley Periodicals, Inc.