The Annals of Probability
- Ann. Probab.
- Volume 32, Number 4 (2004), 2938-2977.
On the scaling of the chemical distance in long-range percolation models
We consider the (unoriented) long-range percolation on ℤd in dimensions d≥1, where distinct sites x,y∈ℤd get connected with probability pxy∈[0,1]. Assuming pxy=|x−y|−s+o(1) as |x−y|→∞, where s>0 and |⋅| is a norm distance on ℤd, 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.
Ann. Probab. Volume 32, Number 4 (2004), 2938-2977.
First available in Project Euclid: 8 February 2005
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35] 82B28: Renormalization group methods [See also 81T17]
Biskup, Marek. On the scaling of the chemical distance in long-range percolation models. Ann. Probab. 32 (2004), no. 4, 2938--2977. doi:10.1214/009117904000000577. https://projecteuclid.org/euclid.aop/1107883343.