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

TL;DR

This paper extends the strict inequality of site and bond percolation critical probabilities to Gilbert's continuum percolation model, demonstrating that in supercritical regimes, the site critical probability exceeds the bond critical probability.

Contribution

It proves that the strict inequality of critical probabilities holds for Gilbert's continuum percolation model in the supercritical phase, extending known results from discrete graphs.

Findings

In Gilbert's continuum percolation, $p_c^{site} > p_c^{bond}$ in the supercritical phase.

The strict inequality holds almost surely on the infinite component.

Results extend to higher-dimensional Euclidean spaces.

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.