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

Massimo Franceschetti; Mathew D. Penrose; Tom Rosoman

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arXiv · 1004.1596

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

Massimo Franceschetti·Mathew D. Penrose·Tom Rosoman

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Abstract

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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Massimo Franceschetti, Mathew D. Penrose, Tom Rosoman. 2010-04-29. Strict inequalities of critical probabilities on Gilbert's continuum percolation graph. https://arxiv.org/abs/1004.1596

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