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

Massimo Franceschetti; Mathew D. Penrose; Tom Rosoman

paper

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

arXiv:1004.1596

Abstract

Any infinite graph has site and bond percolation critical probabilities satisfying $p_{c}^{s i t e} \geq p_{c}^{b o n d}$ . 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 $p_{c}^{s i t e} > p_{c}^{b o n d}$ . This also holds in higher dimensions.

15 Pages, 3 figures. In this version the main result is unchanged, but there are some clarifications in the proof, particularly with regard to Lemma 3.