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.