## Erik Broman (03/10/16)

Anders Claesson, September 28, 2016

Math Colloquium

### Title: Infinite range continuum percolation models

Location: V-158, VRII
Time: Monday 3 October at 15:00

### Abstract:

In the classical Boolean percolation model, one distributes balls in $$R^d$$ in a random, homogeneous way. The density of these balls is controlled by a parameter $$\lambda.$$ Depending on this density, the collection of balls then either form an infinite cluster ($$\lambda$$ large) or consists of only small components ($$\lambda$$ small).

In the talk I will discuss two variants of this model, both which are infinite range. In the first case, the balls are replaced by bi-infinite cylinders with radius 1. We then investigate what the connectivity structure of the resulting set is, and how this depends on $$\lambda$$ as well as on the underlying geometry (Euclidean vs hyperbolic).

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 for every $$y\in R^d$$ we define $$\Psi(y):=\sum_{x\in \eta}l(|x-y|)$$. 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.

The aim is that the talk should be accessible to anyone with a mathematical, but not necessarily probabilistic, background.