SCS

Abstract

For any infinite connected graph, the critical probabilities for bond percolation and for site percolation satisfy the inequality $p_c^{\rm bond} \leq p_c^{\rm site}$. Moreover, this is known to be a strict inequality on a large class of lattices. In this talk, we discuss the extension of the strict inequality to certain {\em random} graphs arising in continuum percolation, including the Gilbert graph in which each point of a homogeneous planar Poisson point process of supercritical intensity $\lambda$ is connected by an edge to every other Poisson point within unit distance. More generally, we consider the random connection model, in which each pair of Poisson points distant at most $r$ apart is connected by an edge with probability $p$, so that the average node degree is $\lambda \pi r^2 p$. Given $r$ and $p$ there is a critical intensity $\lambda_c(p,r)$ above which the graph percolates. As well as the strict inequality $p_c^{\rm bond} < p_c^{\rm site}$ for this graph, we discuss a related result which says that $\lambda_c(r^{-2},r)$ is strictly decreasing in $r $. That is, for a given average node degree it is easier to percolate in a graph with long-range connections than in a graph with only short-range connections.