IDLA aggregates with infinitely many sources and the directed IDLA forest

Abstract

We investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on Z 2 with infinitely many sources that are the points of the vertical axis I (∞) = {0} × 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 Z 2 , based on an IDLA protocol, which is invariant in distribution w.r.t. vertical translations.