text: We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in article amsmath,amsfonts empty . We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x − y|−s+o(1) when |x − y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)Δ+o(1) where Δ−1 := log2(2d s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp r1 Δ+o(1) in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210-227, 2011 (Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.)
  @cite_9: We prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process. We improve earlier results of Gartner and Molchanov and Grimmett and Marstrand and answer affirmatively a conjecture of Kozlov.
  @cite_13: Bounds for the diameter and expansion of the graphs created by long-range percolation on the cycle ℤ Nℤ are given. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 102–111, 2001
  @cite_8: We consider long-range percolation in dimension @math , where distinct sites @math and @math are connected with probability @math . Assuming that @math is translation invariant and that @math with @math , we show that the graph distance is at least linear with the Euclidean distance.
summary: For @math one expects @cite_13 the same behavior as for the original graph. And indeed, the linear asymptotic, has been established by Berger @cite_8 . For the nearest-neighbor percolation case, this statement goes back to the work of Antal and Pisztora @cite_9 .