2003/04/30 by Marek Biskup · 10 citations
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60K35 #msc:82B28. #msc:82B43
paper · pdf · doi:10.1214/009117904000000577
published as Annals of Probability 2004, Vol. 32, No. 4, 2938-2977 · Published at http://dx.doi.org/10.1214/009117904000000577 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2004/10/01 · arxiv created 2005/04/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.