Description
In the Abelian sandpile model, we have a finite connected graph G=(V,E) and a marked vertex s. At every vertex v other than s, we place at most deg(v)-1 sand particles, which then evolve according to the following dynamics. At each time step, we add a sand particle to a randomly chosen vertex. Whenever a vertex v reaches deg(v) particles, it "topples", sending one particle to each of its neighbours. Particles arriving at s are removed from the system.
In joint work with Minwei Sun, we study the stationary distribution of the model on lattice boxes numerically, using exact sampling. In particular, we study the following conjecture: if we add one particle at the origin to a stationary sandpile, then the probability that a vertex x topples is asymptotically c |x|^{2-d-\eta}, for some exponent \eta that is >0 for d=2,3. Our simulation method, using a combination of Wilson's algorithm to generate uniformly distributed spanning trees, and Majumdar and Dhar's bijection with sandpiles, lends itself well to studying some problems even in very high dimensions. We illustrate this with an estimation of the height distribution at the origin in d=32, and compare this to the asymptotic behaviour as d\to\infty.