The diameter of a long range percolation graph

We consider the following long range percolation model: an undirected graph with the node set $\{0,1,\mathrm{.}\mathrm{.}\mathrm{.},N{\}}^d$, has edges $(\text{x},\text{y})$ selected with probability $\approx \beta \mathrm{/}\mathrm{\mid }\mathrm{\mid }\text{x}-\text{y}\mathrm{\mid }{\mathrm{\mid }}^s$ if $\mathrm{\mid }\mathrm{\mid }\text{x}-\text{y}\mathrm{\mid }\mathrm{\mid }>1$, and with probability 1 if $\mathrm{\mid }\mathrm{\mid }\text{x}-\text{y}\mathrm{\mid }\mathrm{\mid }=1$, 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 $\mathrm{\Theta }(\log N\mathrm{/}\log \log N)$ 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.