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 site c > p bond c . 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) 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.

On the critical threshold for continuum AB percolation

“…By scaling (see Proposition 2.11 of [8]) λ c (2r) = r −d λ c (2). The value of λ c (2) is not known analytically, but is well known to be finite for d ≥ 2 [4,8], and explicit bounds are provided in [8].…”

“…Given µ, we use a version of the technique of enhancement to show that there exists a value of λ such that P λ is supercritical for Apercolation (with distance parameter 2r) but becomes subcritical after removal of all the useless particles (a thinning process with only local dependence). The technique of enhancement has previously been applied to one-type continuum percolation in [2,3], and further discussion of enhancement can be found there.…”

“…Proof. Adapting the proof of Lemma 1 in [2], we prove only the first identity since the second is obtained in exactly the same way. We denote by F the σ-algebra generated by (P λ 0 , Q µ ).…”

“…When B R (x) ∩ B c n = ∅, consider a path (x i ) 1≤i≤2R+2 from outside B 2R (x) to x and a path (y i ) 1≤i≤R+2 inside B 2R (x) joining x to B c n . For brevity we omit the details of these cases here (which are similar to the corresponding cases treated in [2]). Lemma 2.2 is proved.…”

Russo's Formula, Uniqueness of the Infinite Cluster, and Continuous Differentiability of Free Energy for Continuum Percolation

“…This corollary gives the formula for the derivative of the probability of a certain kind of event. In fact, one can also get this formula as a special case of Theorem 2.1 of [15]; see also Lemma 1 of [4].…”