Frank Redig

I will give an introduction
to the ``abelian sandpile model" (ASM).
In this model to every vertex i of
a finite subset V of Z^{d} we associate a integer valued
height variable h_{i}>0 (``the number of grains at i").
If h_{i} < 2d+1 the site is called stable. An unstable
configuration is stabilized by a sequence of ``topplings"
consisting of removing 2d grains from unstable
sites and giving one grain to each neighbor. In the ASM dynamics,
one picks a vertex at random, adds a grain and stabilizes
the configuration.
The recurrent configurations of ASM form an abelian group (under pointwise addition and stabilization). They can also be characterized by an algorithm which gives a bijection to the set of rooted spanning trees on V. The effect of adding one grain to a recurrent configuration can be characterized by a sequence of ``waves", each of which can be represented by a two component spanning tree. Spanning trees are related to loop-erased random walk via Wilson's algorithm. If time is left, I will discuss some recent results for the ASM on infinite graphs. |