On the scaling of the chemical distance in long-range percolation models
The paper deals with long-range percolation on \({\mathbb Z}^d\) where two sites \(x\) and \(y\) get connected with probability \(p_{xy}=| x-y| ^{-s+o(1)}\) as \(| x-y| \rightarrow \infty\). 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 \(s\) relative to \(d\) and \(2d\) 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 \(d<s<2d\) and the random graph contains a unique infinite component. The study of percolation in finite boxes is an essential tool for the proof.
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