At last week’s AIM conference on chip-firing I learned about a cool stochastic process which seems very simple, but whose behavior is apparently very hard to prove anything about!
It goes like this. You start with a graph (say a finite connected graph, for simplicity) and your states are functions from the vertices of the graph to the natural numbers, which you think of as specifying the height of a pile of chips (or sand grains, if you want to keep up that metaphor) sitting on top of that vertex. At least one of the vertices is a sink, which means that any chip landing there disappears forever.
Now the nature of the process is that you never want to have more than one chip at a vertex. So every time there are at least two chips at a vertex v, you can pop two of them off; each chip flips a coin and decides whether to topple left or right. And you keep on doing this until there’s at most one chip at each spot. (You might start with more chips than vertices, but the sink helps you slough off the surplus.) This process was apparently first studied by Manna and is sometimes called the Manna sandpile.
More math below the fold!