30.10.2023

Ecatarina Sava-Huss: Abelian Sandpile Markov chains

The Abelian sandpile model on a graph G is a Markov chain whose state space is a subset of the set of functions with integer values defined on the vertices of G. The set of recurrent states of this Markov chain is called the sandpile group and the Abelian sandpile model can be then viewed as a random walk on a finite group. Then it is natural to ask about the stationary distribution and the speed of convergence to stationarity, and how do these quantities depend on the underlying graph . I will report on some recents results on Abelian sandpiles on fractal graphs, and state some open questions concerning the critical exponents for such processes. The talk is based on joint works with Nico Heizmann, Robin Kaiser and Yuwen Wang.

Dirk Erhard: The tube property for the swiss cheese problem

In 2001 Bolthausen, den Hollander and van den Berg obtained the asymptotics of the probability that the volume of a Wiener sausage at time t is smaller than expected by a fixed muliplicative constant. This asymptotics was given by a variational formula and they conjectured that the best strategy to achieve such a large deviation event is for the underlying Brownian motion to behave like a swiss cheese: stay most of the time inside a ball of subdiffusive size, visit most of the points but leave some random holes. They moreover conjectured that to do so the Brownian motion behaves like a Brownian motion in a drift field given by a function of the maximizer of the variational problem.

In this talk I will talk about the corresponding problem for the random walk and will explain that conditioned to having a small range its properly defined empirical measure is indeed close to the maximizer of the above mentioned variational problem.

This is joint work with Julien Poisat.