The bi-dimensional Directed IDLA forest

TL;DR

This paper develops three infinite-IDLA models on ${\mathbb Z}^2$ with sources along the vertical axis, proving invariance under vertical translations, stabilization, mixing, and shape theorems. It introduces a directed infinite-volume IDLA forest $\mathcal{F}_\infty$ built from three aggregates $A_n[\infty]$, $A_n^*[\infty]$, $A_n^\dag[\infty]$, and shows that the forest is stationary, spans ${\mathbb Z}^2$, and comprises a countable union of directed trees rooted on $I(\infty)$. The work overcomes challenges from the loss of Abelian property and long-range dependencies by establishing weak stabilization and robust bounds, enabling a consistent horizontal limit and a well-defined forest. It also provides detailed shape theorems for the aggregates (including Poisson variants) and discusses open questions about the forest’s finer properties and connections to the classical IDLA tree $\mathcal{T}_\infty$. Overall, the results contribute a stationary, translation-invariant framework for directed IDLA-like growth and a new random forest object with potential links to scaling limits and the Brownian web.

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.

Figures (5)

• Figure 1: A realization of $\mathcal{T}_{1500}$.
• Figure 2: A realization of ${{\mathcal{F}}_{40}^\dag[200]}$ observed in ${\mathbb Z}_{30}$.
• Figure 3: A realization of the aggregate $A_{90}[200]\cap {\mathbb Z}_{20}$ based on 90 particles per site $(0,i)$, with $|i|\leq 200$, and intersected by the strip ${\mathbb Z}_{20}$.
• Figure 4: The top left corner depicts a realization of the random forest ${{\mathcal{F}}_{120}^\dag[120]}$ with particles starting from levels $|i|\leq 120$ and during the time interval $[0,120]$, viewed through the strip ${\mathbb Z}_{15}$. The tree of ${{\mathcal{F}}_{120}^\dag[120]}$ containing the origin is in red. A second realization of ${{\mathcal{F}}_{120}^\dag[120]}$ is given on the bottom left corner. The branch passing through $(55,0)$ (red) remains close to the $x$-axis and comes from the source $(0,2)$. A realization of ${{\mathcal{F}}_{15}^\dag[30]}$ is depicted on the right. One can imagine that the vertical edges at the top of ${{\mathcal{F}}_{15}^\dag[30]}$ are due to border effects and will not be present in the limiting forest ${\mathcal{F}}_{15}$.
• Figure 5: Realizations of the forests ${{\mathcal{F}}_{20}^\dag[20]}$ and ${{\mathcal{F}}_{20}^\dag[50]}$, defined on the same time interval $[0,20]$, with different sets of sources, and restricted to the strip ${\mathbb Z}_{20}$, are depicted. The associated aggregates are coupled in the sense that they are based on the same clocks and random walks with level $|i|\leq 20$. In particular, $A^\dag_{20}[20]$ is included in $A^\dag_{20}[50]$. The edges created in both forests by the same particles are depicted in green. The red points are vertices of $A^\dag_{20}[50]\!\setminus\!A^\dag_{20}[20]$. The blue circles represent vertices in $A^\dag_{20}[20]$ (and then also in $A^\dag_{20}[50]$) which are reached by different particles in both aggregates and whose corresponding edges may differ in both forests ${{\mathcal{F}}_{20}^\dag[20]}$ and ${{\mathcal{F}}_{20}^\dag[50]}$. These blue vertices are possible discrepancies generated by chains of changes between forests ${{\mathcal{F}}_{20}^\dag[20]}$ and ${{\mathcal{F}}_{20}^\dag[50]}$.

Theorems & Definitions (30)

• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3: Crossing Lemma
• Lemma 3.4
• Remark 3.5
• Proposition 3.6