  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 $l:(0,\infty) \to (0,\infty)$ to create the
random field $\Psi(y)=\sum_{x\in \eta}l(|x-y|),$ 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 $\Psi_{\geq h}(y)$ containing the
points $y\in {\mathbb R}^d$ such that $\Psi(y)\geq h.$ In the case where $l$
has unbounded support, we give, for any $d\geq 2,$ exact conditions on $l$ for
$\Psi_{\geq h}(y)$ 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 $\Psi_{\geq h}$ when such exists, and we also show that the
so-called percolation function is continuous below the critical value $h_c$.
