Phase Transition and Uniqueness of Levelset Percolation
Artikel i vetenskaplig tidskrift, 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.

Continuous percolation

Continuity of the field




Erik Broman

Göteborgs universitet

Chalmers, Matematiska vetenskaper

R. Meester

Vrije Universiteit Amsterdam

Journal of Statistical Physics

0022-4715 (ISSN) 1572-9613 (eISSN)

Vol. 167 6 1376-1400



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