Strict inequalities of critical probabilities on Gilbert's continuum percolation graph

Details Strict inequalities of critical probabilities on Gilbert's continuum percolation graph Strict inequalities of critical probabilities on Gilbert's continuum percolation graph

2010-04-09

arxiv.org/abs/1004.1596v2

Mathematics

Probability

15 Pages, 3 figures. In this version the main result is unchanged, but there are some clarifications in the proof, particularl

Scientific paper

Any infinite graph has site and bond percolation critical probabilities satisfying $p_c^{site}\geq p_c^{bond}$. The strict version of this inequality holds for many, but not all, infinite graphs. In this paper, the class of graphs for which the strict inequality holds is extended to a continuum percolation model. In Gilbert's graph with supercritical density on the Euclidean plane, there is almost surely a unique infinite connected component. We show that on this component $p_c^{site} > p_c^{bond}$. This also holds in higher dimensions.

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