We study the graph constructed on a Poisson point process in @math dimensions by connecting each point to the @math points nearest to it. This graph a.s. has an infinite cluster if @math where @math , known as the critical value, depends only on the dimension @math . This paper presents an improved upper bound of 188 on the value of @math . We also show that if @math the infinite cluster of @math has an infinite subset of points with the property that the distance along the edges of the graphs between these points is at most a constant multiplicative factor larger than their Euclidean distance. Finally we discuss in detail the relevance of our results to the study of multi-hop wireless sensor networks.
The study of random graphs obtained by applying connection rules on stationary point processes is known as continuum percolation. Meester and Roy's monograph on the subject provides an excellent view of the deep theory that has been developed around this general setting @cite_11 . The @math model was introduced by H "aggstr "om and Meester @cite_13 . They showed that there was a finite critical value, @math for all @math such that an infinite cluster exists in this model. They proved that the infinite cluster was unique and that there was a value @math such that @math for all @math . Teng and Yao gave an upper bound of 213 for @math @cite_15 .
@cite_15 @cite_13 @cite_11
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The percolation phase transition and the mechanism of the emergence of the giant component through the critical scaling window for random graph models, has been a topic of great interest in many dierent communities ranging from statistical physics, combinatorics, computer science, social networks and probability theory. The last few years have witnessed an explosion of models which couple random aggregation rules, that specify how one adds edges to existing congurations, and choice, wherein one selects from a ‘limited’ set of edges at random to use in the conguration. These models exhibit fascinating new phenomenon, ranging from delaying or speeding up the emergence of the giant component, to percolation", where the diameter of the scaling window is several orders of magnitude smaller than that for standard random graph models. While an intense study of such models has ensued, understanding the actual emergence of the giant component and merging dynamics in the critical scaling window has remained impenetrable to a rigorous analysis. In this work we take an important step in the analysis of such models by studying one of the standard examples of such processes, namely the Bohman-Frieze model, and provide rst results on the asymptotic dynamics, through the critical scaling window, that lead to the emergence of the giant component for such models. We identify the scaling window and show that through this window, the component sizes properly rescaled converge to the standard multiplicative coalescent. Proofs hinge on a careful analysis of certain innite-type branching processes with types taking values in the space of RCLL paths, and stochastic analytic techniques to estimate susceptibility functions of the components all the way through the scaling window where these functions explode. Previous approaches for analyzing random graphs at criticality have relied largely on classical breadth-rst search techniques that exploit asymptotic connections with Brownian excursions. For dynamic random graph models evolving via general Markovian rules, such approaches fail and we develop a quite dierent set of tools that can potentially be used for the study of critical dynamics for all bounded size rules. 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. 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 introduce a model of random interlacements made of a countable collection of doubly infinite paths on ℤ d ,d > 3. A nonnegative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder (ℤ Nℤ) d-1 x ℤ by simple random walk, or the set of points visited by simple random walk on the discrete torus (ℤ Nℤ) d at times of order u N d . In particular we study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite connected translation invariant random subset of ℤ d . We introduce a critical value u * such that the vacant set percolates for u u * . Our main results show that u * is finite when d > 3 and strictly positive when d > 7.