Given finite sets A and B in the lattice, the Diaconis-Fulton sum is a random set obtained by starting one particle at every point of their symmetric difference, and two particles at every point of their intersection. Each “extra” particle performs random walk until it reaches an unoccupied site. 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 domains in Euclidean space. The limit is described by focusing on the “odometer”
of the process, which solves a free boundary obstacle problem for the Laplacian. Joint work with Yuval Peres.