Appears in collection : Random walks: applications and interactions / Marches aléatoires: applications et interactions

Internal Dilusion Limited Aggregation is an interacting particle system that describes the growth of a random cluster governed by the boundary harmonic measure seen from an internal point. We propose a new variant of internal DLA driven by critical branching random walks. We prove that, unlike classical internal DLA, this process exhibits a phase transition in the dimension. More precisely, we establish the existence of a spherical shape theorem in dimension d ≥ 3 and the absence of a spherical shape theorem for d ≤ 2. The outer bound requires a new method to deal with the branching feature of our di!usions. Our bounds on the inner and outer worst deviations are of polynomial nature, which we expect to be a feature of this model.