On the critical threshold for continuum AB percolation

TL;DR

This paper establishes a lower bound for the percolation threshold in a bipartite random geometric graph, revealing how it diverges as the parameter approaches a critical value, thus advancing understanding of phase transitions in such models.

Contribution

It provides a new lower bound for the percolation threshold in bipartite continuum percolation models, linking it to the standard Poisson Boolean model's critical point.

Findings

Lower bound for _c() in bipartite models

Threshold diverges as  approaches _c from above

Insights into phase transition behavior in continuum 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.