On the critical threshold for continuum AB percolation

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 $\mu_c(\lambda)\ge c\log(c/(\lambda-\lambda_c))$forany$\lambda>\lambda_c$with$c>0$ a fixed constant. In particular, ${\mu }_c(\lambda )$ tends to infinity when $\lambda$ tends to ${\lambda }_c$ from above.