2004
DOI: 10.1214/009117904000000577
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On the scaling of the chemical distance in long-range percolation models

Abstract: We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\infty}. Our main result is that, for s\in(d,2d), D(x,y)=(\log|x-y|)^{\Delta+o(1)},\qquad … Show more

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Cited by 117 publications
(233 citation statements)
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“…(They also studied the case in which the distance ρ(·, ·) is defined by assuming that the integers are "wrapped" into a cycle, so that ρ(i, j ) is not |j − i| but min(|j − i|, n − |j − i|).) Their work was followed by results of Coppersmith et al [20] and Biskup [13], who obtained sharper bounds in some cases and considered higher-dimensional lattices as well, in which the node set is {1, 2, . .…”
Section: Further Results On Small-world Network and Decentralized Sementioning
confidence: 99%
See 1 more Smart Citation
“…(They also studied the case in which the distance ρ(·, ·) is defined by assuming that the integers are "wrapped" into a cycle, so that ρ(i, j ) is not |j − i| but min(|j − i|, n − |j − i|).) Their work was followed by results of Coppersmith et al [20] and Biskup [13], who obtained sharper bounds in some cases and considered higher-dimensional lattices as well, in which the node set is {1, 2, . .…”
Section: Further Results On Small-world Network and Decentralized Sementioning
confidence: 99%
“…As a result of this work, we know that the diameter of the graph changes qualitatively at the "critical values" α = d and α = 2d. In particular, with high probability, the diameter is constant when α < d (due in essence to a result of [12]), is proportional to log n/ log log n when α = d [20], is polylogarithmic in n when d < α < 2d (with an essentially tight bound provided in [13]), and is lower-bounded by a polynomial in n when α > 2d [11], [20]. The case α = 2d is largely open, and conjectured to have diameter polynomial in n with high probability [11], [13].…”
Section: Further Results On Small-world Network and Decentralized Sementioning
confidence: 99%
“…Here we conjecture: Note that, according to this conjecture, in d = 1, the interval α ∈ (0, 2) of "interesting" exponents is larger than the interval for which an infinite connected component may occur even without the "help" of nearest neighbor connections. On the other hand, in dimensions d ≥ 3, the interval conjectured for stable convergence is strictly smaller than that of "genuine" long-range percolation behavior, as defined, e.g., in terms of the scaling of graph distance with Euclidean distance; cf [4,5,8].…”
Section: B Some Questions and Conjecturesmentioning
confidence: 98%
“…In the presence of SR infection, when no cooperative effects are considered and there is immunization or death after the infection, the epidemic process gives rise to ordinary percolation clusters [16]. The generalization to cases where the dynamical process takes place in lattices where at least some of the infections are LR was considered [17][18][19]. Further interest in the study of LR percolation was also triggered by papers giving some more exact results and its realization on finite graphs [20,21].…”
Section: Introductionmentioning
confidence: 99%