Continuum percolation with steps in the square or the disc
Random Structures & Algorithms
The phase transition in inhomogeneous random graphs
Random Structures & Algorithms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Random Structures & Algorithms
Combinatorics, Probability and Computing
Combinatorics, Probability and Computing
Modeling and connectivity analysis in obstructed wireless ad hoc networks
Proceedings of the 15th ACM international conference on Modeling, analysis and simulation of wireless and mobile systems
Connectivity in obstructed wireless networks: from geometry to percolation
Proceedings of the fourteenth ACM international symposium on Mobile ad hoc networking and computing
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Given ω ≥ 1, let be the graph with vertex set in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus is precisely .) Let pc(ω) be the critical probability for site percolation on . Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.