On the critical threshold for continuum AB percolation

Abstract

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda ,\mu$. For any $\lambda >0$ we consider the percolation threshold ${\mu }_c(\lambda )$ associated to the parameter $\mu$. Denoting by ${\lambda }_c:={\lambda }_c(2r)$ the percolation threshold for the standard Poisson Boolean model with radii $r$, we show the lower bound ${\mu }_c(\lambda )\ge c\log (c/(\lambda -{\lambda }_c))$ for any $\lambda >{\lambda }_c$ with $c>0$ a fixed constant. In particular, ${\mu }_c(\lambda )$ tends to infinity when $\lambda$ tends to ${\lambda }_c$ from above.

12 pages, 3 figures

References in corpus (2)

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