On the scaling of the chemical distance in long-range percolation models

Abstract

We consider the (unoriented) long-range percolation on ℤ^d in dimensions d≥1, where distinct sites x,y∈ℤ^d get connected with probability p_xy∈[0,1]. Assuming p_xy=|x−y|^−s+o(1) as |x−y|→∞, where s>0 and |⋅| is a norm distance on ℤ^d, and supposing that the resulting random graph contains an infinite connected component C_∞, we let D(x,y) be the graph distance between x and y measured on C_∞. Our main result is that, for s∈(d,2d), D(x,y)=(log|x−y|)^Δ+o(1), x,y∈C_∞, |x−y|→∞, where Δ⁻¹ is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x−y|→∞. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of “small-world” phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.