## 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 , ∞) → [ 0 , ∞) to create the random field Ψ (y) = ∑ _{x} _{∈} _{η}l(| x- y|) , where η is a homogeneous Poisson process in 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 _{≥} _{h}(y) containing the points y∈ R^{d} 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 h_{c}.

Original language | English |
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Pages (from-to) | 1376-1400 |

Number of pages | 25 |

Journal | Journal of Statistical Physics |

Volume | 167 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jun 2017 |

## Keywords

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