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.