The driven-dissipative Abelian sandpile model on a finite graph is a Markov chain on configurations of particles situated on the vertices of the graph. At each step, a new particle is added at a random location, and the configuration is "stablised" via a deterministic process, called an "avalanche", defined in terms of simple local rules. The interest is in the power law statistics of avalanche sizes, that in dimensions d=2 and d=3 is largely conjectural. Each avalanche can be decomposed into simpler objects, called waves, that can be studied via a bijection with variants of the uniform spanning tree. We show two results about waves in the 2D model. Let D be a bounded simply connected domain, a>0 a small lattice spacing, and D(a) a lattice approximation of D. Let o be a point in D, and let W be the set of vertices in a uniformly random wave belonging to an avalanche initiated at o in the model on D(a).
(1) We have that log (in-radius(W)) / log(1/a) converges to a uniform random variable as a->0.
(2) Conditioned on W containing another point z in D, the probability that dist(W,D^c) < b is of order 1/log(1/b), uniformly in 0<a<a_0 for some a_0.