The bi-dimensional Directed IDLA forest

Nicolas Chenavier; David Coupier; Arnaud Rousselle

doi:10.1214/22-AAP1865

Abstract

We investigate three types of internal diffusion limited aggregation (IDLA) models. These models are based on simple random walks on ${\mathbb{Z}^{2}}$ with infinitely many sources that are the points of the vertical axis $I(\infty )=\{0\}\times \mathbb{Z}$ . 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 ${\mathbb{Z}^{2}}$ , 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.

Acknowledgements

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

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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

Subjects:

Primary: 05C80 , 60K35 , 82C24

Secondary: 60G50 , 82B41

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

Abstract

Funding Statement

Acknowledgements

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