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

We consider the (unoriented) long-range percolation on Z(d) in dimensions d greater than or equal to 1, where distinct sites x,y is an element of Z(d) get connected with probability p(xy) is an element of [0, 1]. Assuming p(xy) = \x - y\(-s+o(1)) as \x - y\ --> infinity, where s > 0 and \ (.) \ is a norm distance on Z(d), and supposing that the resulting random graph contains an infinite connected component Cinfinity, we let D(x, y) be the graph distance between x and y measured on Cinfinity. Our main result is that, for s is an element of (d, 2d), D(x, y) = (log \x - y\)Delta+o(1), x, y is an element of Cinfinity, \x - y\ --> infinity where Delta(-1) is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as \x -y\ --> infinity. 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.