Random Walks on Sandpile Groups

Опубликовано 12 июля 2016, 22:50

The sandpile group of a finite graph is an abelian group that is defined using the graph Laplacian. I will describe a natural random walk on this group. The main questions are: what is the mixing time of the sandpile random walk, and how is it affected by the geometry of the underlying graph? These questions can sometimes be answered even if the actual group is unknown. In particular, the spectral gaps of the sandpile walk and the simple random walk on the underlying graph exhibit a surprising inverse relationship. I will discuss techniques for analyzing the spectral gap and mixing time, including a lattice invariant called the ``smoothing parameter'' that proves cutoff for the sandpile walk on the complete graph. This is joint work with Lionel Levine and John Pike.