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

Massimo Franceschetti; Mathew D. Penrose; Tom Rosoman

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

Probability 2010-04-30 v2

Authors: Massimo Franceschetti , Mathew D. Penrose , Tom Rosoman

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.

Keywords

continuum percolation Gilbert's graph critical probability site percolation bond percolation strict inequality

Cite

@article{arxiv.1004.1596,
  title  = {Strict inequalities of critical probabilities on Gilbert's continuum percolation graph},
  author = {Massimo Franceschetti and Mathew D. Penrose and Tom Rosoman},
  journal= {arXiv preprint arXiv:1004.1596},
  year   = {2010}
}

Comments

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.

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

Abstract

Keywords

Cite

Comments

Related papers