Erik Broman: Continuum percolation models with infinite range

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 \(l:(0,\infty) \to(0,\infty)\) 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 l 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.