The Annals of Probability

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

Marek Biskup

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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=|xy|s+o(1) as |xy|→∞, 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|xy|)Δ+o(1),  x,yC, |xy|→∞, where Δ−1 is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |xy|→∞. 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.

Article information

Ann. Probab. Volume 32, Number 4 (2004), 2938-2977.

First available: 8 February 2005

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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]

Long-range percolation chemical distance renormalization small-world phenomena


Biskup, Marek. On the scaling of the chemical distance in long-range percolation models. The Annals of Probability 32 (2004), no. 4, 2938--2977. doi:10.1214/009117904000000577.

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