The paper deals with long-range percolation on
where two sites
get connected with probability
. The interest focuses on the scaling of the graph distance (or chemical distance) between two remote sites. The introduction gives an overview of previous works concerning the five distinct regimes marked by the position of
and discusses the relationship with the so-called “small-world” phenomena. The main result of the paper is a proof of a polylogarithmic estimation on the asymptotic behavior of the graph distance when
and the random graph contains a unique infinite component. The study of percolation in finite boxes is an essential tool for the proof.