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

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Any infinite graph has site and bond percolation critical probabilities satisfying pcsite≥ pcbond. 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 pcsite > pcbond. This also holds in higher dimensions.