Dirk Erhard

The goal of this talk is two-fold. Firstly, I want to explain some typical questions that we study in the theory of probability. Secondly, I want to show how the analyses of variational problems and topological considerations naturally enter the game. To that end I will focus on a simple example, namely that of a simple random walk. A simple random walk is a particle on Z^d that at each unit of time jumps to one of its 2d nearest neighbors uniformly at random. This is a very well understood model. However, this changes dramatically once we force it to visit fewer points than usual. In that case its behaviour is conjectured to behave like a swiss cheese: stay in a relatively small ball, visit most of the points but leave some random holes of random sizes. I will explain what is known and what is not known about that conjecture.  This talk should be suitable for mathematicians with little to no knowledge in probability theory.

This is joint work with Julien Poisat (Paris Dauphine).