We plan to study a model of interaction banching random walks (BRW) on the euclidean lattice Zd, where the walk interact through aggregation in the so-called internal Diffusion Limited Aggregation, (internal DLA).

Let us first recall what is the process when we consider simple random walk, rather than BRW. Internal DLA is a random aggregation process in which the growth of discrete clusters is governed by the harmonic measure seen from an internal point. That is, a simple random walk is released from inside the cluster, and its exit location is added to it. The asymptotic shape of IDLA on Zd starting from a single seed has long been known to be a Euclidean ball, with very small fluctuations. We would like to discuss during this week at CIRM a natural variant of IDLA, namely Branching IDLA, in which the particles that drive the process perform critical branching random walks rather than simple random walks. We have discovered that BIDLA has a strikingly different phenomenology, namely a phase transition from macroscopic fluctuations in low dimension to the existence of a shape theorem in higher dimension. Joelle Ishak is a graduate student at Villetaneuse, and Vittoria Silvestri a junior faculty at Roma I, when Amine Asselah is a senior faculty at Créteil.