Erik Broman (03/10/16)

Anders Claesson, september 28, 2016

Málstofa í stærðfræði

Fyrirlesari: Erik Broman, Chalmers University of Technology and Gothenburg University

Titill: Infinite range continuum percolation models

Staðsetning: V-158, VRII
Tími: Mánudaginn 3. október kl. 15:00

Ágrip:

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.