Existence of phase transition for heavy-tailed continuum percolation

Let (Xn, rn)n≥1 be a marked Poisson process with intensity λ in Rd, d ≥ 2. The marks (rn) are radii of closed Euclidean balls centered at the points (Xn). Two points Xi and Xj of the Poisson process X are adjacent, Xi ∼ Xj , if D(Xi, ri) ∩ D(Xj , rj) 6= ∅, where D(x,R) = {y ∈ Rd : ||x− y||2 ≤ R}. We say that x, y ∈ Rd are connected, x↔ y, if there are Xi1 , . . . , Xil ∈ X such that x ∈ D(Xi1 , ri1), y ∈ D(Xil , ril) and Xik ∼ Xik+1 for all 1 ≤ k < l. For x ∈ Rd, let I = {i : x ↔ Xi} and Cx = ∪i∈ID(Xi, ri). Set Cx is called the cluster at x. The number of elements in I is called the size of the cluster, and is denoted |Cx|. We write Pλ for the probability measure associated with X. Continuum percolation was introduced by Gilbert [7] as a model of random network in communication theory. It has recently attracted a lot of attention because of its importance in various applications including wireless networks, sensor networks etc (see [6] and many references therein). For the physical applications of continuum percolation we refer the reader to [13]. The first rigorous analysis of the model is given in [9, 15, 16]. Basic methods for continuum percolation are developed in [2, 9, 14, 15, 16]. The uniqueness of unbounded occupied and vacant components is proved in [11]. The principal reference for continuum percolation is [10]. Similarities between continuum and lattice percolation were noted by Gilbert [7]. However the effect of unbounded radii on the properties of a cluster makes continuum percolation essentially different from the lattice one. The difference was noted in [9]. It is known that in the case of site or bond percolation on Zd, the critical probability at which percolation takes place is often the same as the probability at which mean cluster size becomes infinite [1, 12]. Hall [9] showed that for continuum percolation, the critical intensities at which cluster size and mean cluster size become infinite are not necessarily the same. More precisely (see [9, 10] for the proof and [5] for a more general result),