Phase transition and uniqueness of levelset percolation

Abstract

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 $Ψ(y)=\sum_{x\in η}l(|x-y|),$ where $η$ is a homogeneous Poisson process in ${\mathbb R}^d.$ 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 $Ψ_{\geq h}(y)$ containing the points $y\in {\mathbb R}^d$ such that $Ψ(y)\geq h.$ In the case where $l$ has unbounded support, we give, for any $d\geq 2,$ exact conditions on $l$ for $Ψ_{\geq h}(y)$ to have a percolative phase transition as a function of $h.$ We also prove that when $l$ is continuous then so is $Ψ$ almost surely. Moreover, in this case and for $d=2,$ we prove uniqueness of the infinite component of $Ψ_{\geq h}$ when such exists, and we also show that the so-called percolation function is continuous below the critical value $h_c$.

25 pages