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In the continuum percolation problem, objects of given shape are placed according to a k-dimensional Poisson process. Two objects are neighbors if they overlap. The percolation is said to occur if with a positive probability a random object belongs to an infinite cluster, which exists only when the density of objects is above the so-called critical percolation threshold. Among many other fields, continuum percolation theory has been applied to study wireless networks. A wealth of results exist for two-dimensional percolation since the early work of Broadbent and Hammersley. In the basic continuum model, the objects are permeable discs, for which the first estimate of the critical percolation threshold was given by Gilbert in (in the context of wireless multi-hop networks).
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