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

By Marek Tomasz Biskup. 2003

138 citations recorded by the source corpus.

We consider the (unoriented) long-range percolation on Z d in dimensions d ≥ 1, where distinct sites x, y ∈ Z d get connected with probability p xy ∈ [0, 1]. Assuming p xy = |x - y| -s+o(1) as |x - y| → ∞, where s > 0 and |.| is a norm distance on Z d , 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f5641017d8e05e9e8428e61f09010b475d517311

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Scale-free percolation (2011, unknown): “Furthermore, conditionally on that 0 and x are in the inﬁnite component, Biskup [9] shows that d( 0 ,x) grows like ( log | x | ) Δ for an explicit Δ > 1 when α ∈ (d, 2 d) and Berger [5] shows that it grows at least like | x | when α > 2 d .”
Quenched invariance principle for simple random walk on percolation clusters (2005, background): “, in terms of the scaling of graph distance with Euclidean distance; cf [4, 5, 8].”
Critical phenomena in complex networks (2007, background): “In marked contrast to this, in Kleinberg’s networks (as well as in the long-range percolation problem), there exists a sharp transition between these geometries at some special value of the control parameter—exponent α (Benjamini and Berger, 2001; Biskup, 2004; Martel and Nguyen, 2004).”