The Rotor-router model and Diaconis-Fulton Addition

Given two sets A and B in the lattice, the Diaconis-Fulton sum is a random set obtained by putting one particle in every point of the symmetric difference, and two particles in every point of the intersection, of A and B. Each 'extra particle' performs random walk until it reach an unoccupied site, where it settles. The law of the resulting random occupied set A+B does not depend on the order of the walks. We find the (deterministic) scaling limit of the sums A+B when A and B consist of the lattice points in some overlapping planar domains. The limit is described by focusing on the 'odometer' of the process, which solves a free boundary obstacle problem for the Laplacian.

The same scaling limit is obtained when the random walks are replaced by deterministic rotor walks, as proposed by Jim Propp. In particular, when a singleton at the origin is added to itself n times Internal Diffusion-limited aggregation (IDLA) arises; Lawler, Bramson and Griffeath (1992) proved the limit shape for IDLA is a disk and we prove the analogous result for rotor-router aggregation. (Joint work with Lionel Levine.)