Graph diameter in long-range percolation

Abstract

We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an edge between $x$ and $y$ is added with probability decaying with the Euclidean distance as $|x-y|^{-s+o(1)}$ when $|x-y|\to\infty$. For $s\in(d,2d)$ we show that the graph diameter for the graph reduced to a box of side $L$ scales like $(\log L)^{\Delta+o(1)}$ where $\Delta^{-1}:=\log_2(2d/s)$. In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance $L$. We also show that a ball of radius $r$ in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius $\exp\{r^{1/\Delta+o(1)}\}$ in the Euclidean metric.