June 2023 The bi-dimensional Directed IDLA forest
Nicolas Chenavier, David Coupier, Arnaud Rousselle
Author Affiliations +
Ann. Appl. Probab. 33(3): 2247-2290 (June 2023). DOI: 10.1214/22-AAP1865


We investigate three types of internal diffusion limited aggregation (IDLA) models. These models are based on simple random walks on Z2 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 Z2, based on an IDLA protocol, which is invariant in distribution w.r.t. vertical translations.

Funding Statement

This work was partially supported by the French ANR grant ASPAG (ANR-17-CE40-0017), by the French RT GeoSto (RT-3477), and by the French PEPS-JCJC 2019.


We thank our Ph.D. student, Keenan Penner, for pointing us a mistake in our paper and two anonymous referees for suggestions (in particular for pointing us the references [34, 35]) and improvements of the manuscript.


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Nicolas Chenavier. David Coupier. Arnaud Rousselle. "The bi-dimensional Directed IDLA forest." Ann. Appl. Probab. 33 (3) 2247 - 2290, June 2023. https://doi.org/10.1214/22-AAP1865


Received: 1 August 2021; Revised: 1 March 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583670
zbMATH: 1511.60142
Digital Object Identifier: 10.1214/22-AAP1865

Primary: 05C80 , 60K35 , 82C24
Secondary: 60G50 , 82B41

Keywords: Cluster growth , internal diffusion limited aggregation , random trees and forests , Random walks , shape theorems

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 3 • June 2023
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