Tom Hutchcroft (Cambridge): Scaling exponents for high-dimensional spanning forests and sandpiles

The uniform spanning forests (USFs) of infinite an infinite graph $G$ are defined as infinite volume limits of uniform spanning trees on finite subgraphs of $G$. In this talk, I will describe how we use a new way of sampling the USF using the random interlacement process to compute various critical exponents governing the large-scale geometry of trees in the forest in a wide variety of "high-dimensional" graphs, including $Z^d$ for $d \geq 5$ and every bounded degree nonamenable graph. I will then sketch how this allows us to compute related exponents describing the geometry of avalanches in the Abelian sandpile model on the same class of graphs.