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∈ ηl(|x-y|), where η is a homogeneous Poisson process in Rᵈ. 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∈ Rᵈ 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 hc .