Phase Transition and Uniqueness of Levelset Percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function l: (0 , ∞) → [ 0 , ∞) to create the random field Ψ (y) = ∑ _{x ∈ η}l(| x- y|) , where η is a homogeneous Poisson process in R^d. The field Ψ is then a random potential field with infinite range dependencies whenever the support of the function l is unbounded. In particular, we study the level sets Ψ _{≥ h}(y) containing the points y∈ R^d such that Ψ (y) ≥ h. In the case where l has unbounded support, we give, for any d≥ 2 , a necessary and sufficient condition on l for Ψ _{≥ h}(y) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is Ψ almost surely. Moreover, in this case and for d= 2 , we prove uniqueness of the infinite component of Ψ _{≥ h} when such exists, and we also show that the so-called percolation function is continuous below the critical value h_c.