Strict inequalities of critical values in continuum percolation

Abstract

We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\mathrm{site}}>p_c^{\mathrm{bond}}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) {\em strictly} reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.

38 pages, 8 figures

References in corpus (1)

• Continuum percolation with steps in an annulus

Cited by in corpus (4)

• Clustering and percolation of point processes
• On maximal hard-core thinnings of stationary particle processes
• On the critical threshold for continuum AB percolation
• The work of Lucio Russo on percolation