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October 2004 On the scaling of the chemical distance in long-range percolation models
Marek Biskup
Ann. Probab. 32(4): 2938-2977 (October 2004). DOI: 10.1214/009117904000000577

Abstract

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.

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Marek Biskup. "On the scaling of the chemical distance in long-range percolation models." Ann. Probab. 32 (4) 2938 - 2977, October 2004. https://doi.org/10.1214/009117904000000577

Information

Published: October 2004
First available in Project Euclid: 8 February 2005

zbMATH: 1072.60084
MathSciNet: MR2094435
Digital Object Identifier: 10.1214/009117904000000577

Subjects:
Primary: 60K35
Secondary: 82B28 , 82B43

Keywords: Chemical distance , Long-range percolation , renormalization , small-world phenomena

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 4 • October 2004
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