## The Rotor-router model and Diaconis-Fulton Addition

- Date: 11/27/2006

Yuval Peres (Microsoft Research)

University of British Columbia

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.)

Probability Seminar 2006