Abstract: Planar random growth processes occur widely in the physical world. One of the most well-known, yet notoriously difficult, examples is diffusion-limited aggregation (DLA) which models mineral deposition. This process is usually initiated from a cluster containing a single "seed" particle, which successive particles then attach themselves to. However, physicists have also studied DLA seeded on a line segment. One approach to mathematically modelling planar random growth seeded from a single particle is to take the seed particle to be the unit disk and to represent the randomly growing clusters as compositions of conformal mappings of the exterior unit disk. In 1998, Hastings and Levitov proposed a family of models using this approach, which includes a version of DLA. In this talk I will define a stationary version of the Hastings-Levitov model by composing conformal mappings in the upper half-plane. This is proposed as a candidate for off-lattice DLA seeded on the line. We analytically derive various properties of this model and show that they agree with numerical experiments for DLA in the physics literature.   This talk is based on joint work with Noam Berger and Eviatar Procaccia.