Phase Transition and Uniqueness of Levelset Percolation
Journal article, 2017
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 ? (y) = ? x ? ?l(| x- y|) , where ? is a homogeneous Poisson process in Rd. The field ? 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 ? ? h(y) containing the points y? Rd such that ? (y) ? h. In the case where l has unbounded support, we give, for any d? 2 , a necessary and sufficient condition on l for ? ? h(y) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is ? almost surely. Moreover, in this case and for d= 2 , we prove uniqueness of the infinite component of ? ? h when such exists, and we also show that the so-called percolation function is continuous below the critical value hc.
Continuity of the field