Phase Transition and Uniqueness of Levelset Percolation

Erik Broman, Ronald Meester*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


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 Ψ (y) = ∑ x ηl(| x- y|) , where η is a homogeneous Poisson process in Rd. 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 h(y) containing the points y∈ Rd such that Ψ (y) ≥ h. In the case where l has unbounded support, we give, for any d≥ 2 , a necessary and sufficient condition on l for Ψ 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 h when such exists, and we also show that the so-called percolation function is continuous below the critical value hc.

Original languageEnglish
Pages (from-to)1376-1400
Number of pages25
JournalJournal of Statistical Physics
Issue number6
Publication statusPublished - 1 Jun 2017


  • Continuity of the field
  • Continuous percolation
  • Phase-transition
  • Uniqueness


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