Construction of ergodic IDLA forests in $\mathbb{Z}^d$

TL;DR

This paper establishes the existence of ergodic infinite-volume IDLA forests in multidimensional integer lattices, using a multi-source protocol and percolation techniques, despite the lack of Abelian property.

Contribution

It introduces a new multi-source IDLA forest model in $\

Findings

Existence of infinite-volume IDLA forests in $\

Ergodicity of the constructed IDLA forests

Use of percolation tools to prove stabilization

Abstract

We prove the existence of infinite-volume IDLA forests in $\mathbb{Z}^d$ , with $d \geq 2$, based on a multi-source IDLA protocol. Unlike IDLA aggregates, the laws of the IDLA forests studied here depend on the trajectories of particles, and then do not satisfy the famous Abelian property. Their existence is due to a stabilization result (Theorem 1.1, our main result) that we establish using percolation tools. Although the sources are infinitely many, we also prove that each of them play the same role in the building procedure, which results in an ergodicity property for the IDLA forests (Theorem 1.2).