Tid: 21 mars 2016 kl 15.15-16.15.

Föredragshållare: Erik Broman (Uppsala Universitet)

Titel: Continuum percolation models with infinite range

Abstract In the classical Boolean percolation model, one starts with a homogeneous Poisson process in R^d with intensity u>0, and around each point one places a ball of radius 1. In the talk I will discuss two variants of this model, both which are of infinite range. Firstly, we will consider the so-called Poisson cylinder model, in which the balls are replaced by bi-infinite cylinders of radius 1. We then investigate whether the resulting collection of cylinders is connected, and if so, what the diameter of this set is. In particular I will compare results between Euclidean and hyperbolic geometry. In the second case, we replace the balls with attenuation functions. That is, we let l : (0,infty) -> (0,infty) be some non-increasing function, and then define the random field Psi by letting Psi(y)=sum l(|x-y|), where we sum over all x in the Poisson process. We study the level sets Psi_\geq h which is simply the set of points where the random field Psi is larger than or equal to h: We determine for which functions I this model has a non-trivial phase transition in h: In addition, we will discuss some classical results and whether these can be transferred to this setting. Please note: There will be some overlap with my talk during the Nordic Congress of Mathematicians, but here I will go into more detail and present additional results.

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