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.
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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. We consider critical site percolation on the triangular lattice, that is, we choose @math or 1 with probability 1 2 each, independently for all vertices @math of the triangular lattice. We say that a word @math is seen in the percolation configuration if there exists a selfavoiding path @math on the triangular lattice with @math . We prove that with probability 1 "almost all" words, as well as all periodic words, except the two words @math and @math , are seen. "Almost all" words here means almost all with respect to the measure @math under which the @math are i.i.d. with @math (for an arbitrary @math ). We study the directed polymer model in dimension @math when the environment is heavy-tailed, with a decay exponent @math . We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature @math vanishes as the size of the system n goes to infinity. When @math , we show that all possible transversal fluctuations @math can be achieved by tuning properly @math , allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when @math , we show that there are only two regimes: the transversal fluctuations are either @math or @math . As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP]. We study the number @math of open paths of length @math in supercritical oriented percolation on @math , with @math . We prove that on the percolation event @math , @math almost surely converges to a positive deterministic constant. We also study the existence of directional limits. The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any @math and for any exponent @math giving the rate of decay of the percolation process, we show that the return probability decays like @math up to logarithmic corrections, where @math denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest, the main result is needed for a companion paper studying the scaling limit of simple random walk on the infinite cluster. Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on @math with i.i.d. exponential weights on the vertices. Fix two points @math and @math for some @math , and consider the maximal paths @math and @math starting at @math and @math respectively to the point @math for @math . Our object of study is the point of coalescence, i.e., the point @math with smallest @math . We establish that the distance to coalescence @math scales as @math , by showing the upper tail bound @math for some @math . We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction @math starting from @math and @math , we establish the optimal tail estimate @math , for the point of coalescence @math . This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound. Consider a Poisson process X in Rd with density 1. We connect each point of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolation occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolation model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ball. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general properties of percolation models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). 0 1996 John Wiley & Sons, Inc. Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability @math , independently of each other, and a deterministic spreading rule with a fixed parameter @math : if a vacant site has at least @math occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on @math ; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of @math for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than @math , then the critical probability is 1, while it is @math on the @math -ary tree. A related result is that in any rooted tree @math there is a way of erasing @math children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree @math has branching number @math . We also prove that on any @math -regular non-amenable graph, the critical probability for the @math -rule is strictly positive. We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: @math , the fraction of vertices initially activated, and @math , the fraction of undamaged vertices in the graph. We observe two transitions: the giant active component appears continuously at a first threshold. There may also be a second, discontinuous, hybrid transition at a higher threshold. Avalanches of activations increase in size as this second critical point is approached, finally diverging at this threshold. We describe the existence of a special critical point at which this second transition first appears. In networks with degree distributions whose second moment diverges (but whose first moment does not), we find a qualitatively different behavior. In this case the giant active component appears for any @math and @math , and the discontinuous transition is absent. This means that the giant active component is robust to damage, and also is very easily activated. We also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold. Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. We prove the existence of an asymptotic phase transition threshold on community detectability for the spectral modularity method [M. E. J. Newman, Phys. Rev. E 74, 036104 (2006) and Proc. National Academy of Sciences. 103, 8577 (2006)] under a stochastic block model. The phase transition on community detectability occurs as the inter-community edge connection probability @math grows. This phase transition separates a sub-critical regime of small @math , where modularity-based community detection successfully identifies the communities, from a super-critical regime of large @math where successful community detection is impossible. We show that, as the community sizes become large, the asymptotic phase transition threshold @math is equal to @math , where @math is the within-community edge connection probability. Thus the phase transition threshold is universal in the sense that it does not depend on the ratio of community sizes. The universal phase transition phenomenon is validated by simulations for moderately sized communities. Using the derived expression for the phase transition threshold we propose an empirical method for estimating this threshold from real-world data. Many complex networks in nature have directed links, a property that affects the network’s navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in and out degree distributions. We derive a phase diagram that indicates the existence of three regimes, determined by the values of the degree exponents. In the first regime we regain the known directed percolation mean field exponents. In contrast, the second and third regimes are characterized by anomalous exponents, which we calculate analytically. In the third regime the network is resilient to random dilution, i.e., the percolation threshold is pc!1. We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate @math ( @math ) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent @math , then, as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of @math and @math . The picture is similar with multiple starting points for the infections. We study signal-to-interference plus noise ratio (SINR) percolation for Cox point processes, i.e., Poisson point processes with a random intensity measure. SINR percolation was first studied by in the case of a two-dimensional Poisson point process. It is a version of continuum percolation where the connection between two points depends on the locations of all points of the point process. Continuum percolation for Cox point processes was recently studied by Hirsch, Jahnel and Cali. We study the SINR graph model for a stationary Cox point process in two or higher dimensions. We show that under suitable moment or boundedness conditions on the path-loss function and the intensity measure, this graph has an infinite connected component if the spatial density of points is large enough and the interferences are sufficiently reduced (without vanishing). This holds in all dimensions larger than 1 if the intensity measure is asymptotically essentially connected, and in two dimensions also if the intensity measure is only stabilizing but the connection radius is large. A prominent example of the intensity measure is the two-dimensional Poisson-Voronoi tessellation. We show that its total edge length in a given square has some exponential moments. We conclude that its SINR graph has an infinite cluster if the path-loss function is bounded and has a power-law decay of exponent at least 3. 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. A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the @math -dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent @math of the degree distribution and the rate @math at which the connection probability decreases with the distance of two vertices. We show that the network is robust if @math . In the case of one-dimensional space we also show that the network is not robust if @math . This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.