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_7: We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d= 1,2, where x and y are connected with probability ( ). We show that if d s>2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d≥& 2 and d>s>2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network.
  @cite_1: Consider a one-dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi±j,j≧1. Ifpj is fixed forj≧2 and ( j )j2pj>1, then (unoriented) percolation occurs forp1 sufficiently close to 1. This result, analogous to the existence of spontaneous magnetization in long range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur forp1 close to 1 if ( j )jspj>0 for somes<2. Analogous results are valid for one-dimensional site-bond percolation models.
  @cite_3: We consider one dimensional percolation models for which the occupation probability of a bond −Kx,y, has a slow power decay as a function of the bond's length. For independent models — and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bondsKx,y≦p =1. This dichotomy resembles one for the magnetization in 1 |x−y|2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.
  @cite_2: The problem of long-range percolation in one dimension is proposed. The authors consider a one-dimensional bond percolation system with bonds connecting an infinite number of neighbours where the occupation probability for the nth nearest-neighbour bond pn varies as p1 ns. Using the transfer-matrix method, they find that when s>2 only the short-range percolation exists; namely the system percolates only when p1=1. A transition to long-range percolation is found at s=2 where the percolation threshold drops suddenly from the short-range value p1c=1 to the long-range value p1c=0.
  @cite_10: We rigorously establish the existence of an intermediate ordered phase in one-dimensional 1 |x−y|2 percolation, Ising and Potts models. The Ising model truncated two-point function has a power law decay exponent θ which ranges from its low (and high) temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary. Similar results are obtained for percolation and Potts models.
summary: Long-range percolation, of which our model is an example, originated in the mathema -ti -cal-physics literature as a model that exhibits a phase transition even in spatial dimension one (e.g., Newman and Schulman @cite_1 , Schulman @cite_2 , Aizenman and Newman @cite_3 , Imbrie and Newman @cite_10 ). It soon became clear that @math and @math are two distinguished values; for @math the model is essentially mean-field (or complete-graph) alike, for @math the behavior is more or less as for the nearest-neighbor percolation. The regime @math turned out to be quite interesting; indeed, it is the only general class of percolation models with Euclidean (or amenable) geometry where one can prove absence of percolation at the percolation threshold (Berger @cite_7 ). In all dimensions, the model with @math has a natural continuum scaling limit.