The diameter of a long range percolation graph

Abstract

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 $\approx \beta/||\x-\y||^s$ if $||\x-\y||>1$, and with probability 1 if $||\x-\y||=1$, for some parameters $\beta,s>0$. This model was introduced by Benjamini and Berger, 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 $0<\eta_1<\eta_2<1$, it is at most $N^{\eta_2}$, when $s=2d$ and is at least $N^{\eta_1}$ when $d=1,s=2,\beta<1$ or $s>2d$. 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 by Berger and Benjamini.