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,infty ) rightarrow [0,infty )) to create the random field (Psi (y)=sum _{xin eta }l(|x-y|),) where (eta ) is a homogeneous Poisson process in ({mathbb {R}}^d.) The field (Psi ) 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 (Psi _{ge h}(y)) containing the points (yin {mathbb {R}}^d) such that (Psi (y)ge h.) In the case where l has unbounded support, we give, for any (dge 2,) a necessary and sufficient condition on l for (Psi _{ge h}(y)) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is (Psi ) almost surely. Moreover, in this case and for (d=2,) we prove uniqueness of the infinite component of (Psi _{ge h}) when such exists, and we also show that the so-called percolation function is continuous below the critical value (h_c).