The Rotor-Router Shape is Spherical

In the two-dimensional rotor-router walk (defined by Jim Propp and presented beautifully in [4]), the first time a particle leaves a site x it departs east; the next time this or another particle leaves x it departs south; the next departure is west, then north, then east again, etc. More generally, in any dimension d ≥ 1, for each site x ∈ Z fix a cyclic ordering of its 2d neighbors, and require successive departures from x to follow this ordering. In rotor-router aggregation, we start with n particles at the origin; each particle in turn performs rotor-router walk until it reaches an unoccupied site. Let An denote the shape obtained from rotor-router aggregation of n particles in Z; for example, in Z with the ordering of directions as above, the sequence will be begin A1 = {0}, A2 = {0, (1, 0)}, A3 = {0, (1, 0), (0,−1)}, etc. As noted in [4], simulations in two dimensions indicated that An is close to a ball, but there was no theorem explaining this phenomenon. Order the points in the lattice Z according to increasing distance from the origin, and let Bn consist of the first n points in this ordering; we call Bn the lattice ball of cardinality n. In this letter we outline a proof that for all d, the rotor-router shape An in Z d is indeed close to a ball, in the sense that