On the Scaling of the Chemical Distance in Long-range Percolation Models

@inproceedings{Biskup1983OnTS,
  title={On the Scaling of the Chemical Distance in Long-range Percolation Models},
  author={Marek Biskup},
  year={1983}
}
We consider the (unoriented) long-range percolation on Z in dimensions d ≥ 1, where distinct sites x, y ∈ Z 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 Z, 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|), x, y ∈ C∞, |x− y| →∞, where ∆ is… CONTINUE READING
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