The bi-dimensional Directed IDLA forest

Abstract

We investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on ${\mathbf{Z}}^2$ with infinitely many sources that are the points of the vertical axis $I(\infty )=\{0\}\times \mathbf{Z}$. Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t. the horizontal direction) random forest spanning ${\mathbf{Z}}^2$, based on an IDLA protocol, which is invariant in distribution w.r.t. vertical translations.

40 pages, 7 figures, corrected version