The bi-dimensional Directed IDLA forest

TL;DR

This paper studies three IDLA models on the 2D integer lattice with sources on the vertical axis, establishing properties like stationarity and shape, and introduces a new directed random forest invariant under vertical shifts.

Contribution

It introduces a novel directed IDLA-based random forest on  with invariance properties and analyzes its fundamental characteristics.

Findings

Models exhibit stationarity and mixing properties.

Shape theorems describe the growth and structure.

New directed forest is invariant under vertical translations.

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.