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.)
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.
@cite_7 @cite_1 @cite_3 @cite_2 @cite_10
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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. We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters ( ) and ( ). The non-negative parameter ( ) measures the amount of loops, and ( ) plays the role of killing on vertices penalizing ( ( >0 )) or favoring ( ( <0 )) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed ( ) and large enough ( ), there exists an infinite cluster in the loop percolation on ( Z ^d ). In the present article, we show a non-trivial phase transition on the integer lattice ( Z ^d ) ( (d 3 )) for ( =0 ). More precisely, we show that there is no loop percolation for ( =0 ) and ( ) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of ( ), namely, for ( =0 ) and any sub-critical value of ( ), the probability of one-arm event decays at most polynomially. For (d 5 ), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For ( ) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For (d=3 ) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity ( ). 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. We generalize the standard site percolation model on the @math -dimensional lattice to a model on random tessellations of @math . We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument burton1989density , develop two frameworks that imply the non-triviality of the phase transition and show that large classes of random tessellations fit into one of these frameworks. Our focus is on a very general approach that goes well beyond the typical Poisson driven models. The most interesting examples might be Voronoi tessellations induced by determinantal processes or certain classes of Gibbs processes introduced in schreiber2013 . In a second paper we will investigate first passage percolation on random tessellations.