Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

One-dimensional long-range diffusion-limited aggregation III – The limit aggregate

Gideon Amir

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In this paper we study the structure of the limit aggregate $A_{\infty}=\bigcup_{n\geq0}A_{n}$ of the one-dimensional long range diffusion limited aggregation process defined in (Ann. Probab. 44 (2016) 3546–3579). We show (under some regularity conditions) that for walks with finite third moment $A_{\infty}$ has renewal structure and positive density, while for walks with finite variance the renewal structure no longer exists and $A_{\infty}$ has 0 density. We define a tree structure on the aggregates and show some results on the degrees and number of ends of these random trees. We introduce a new “harmonic competition” model where different colours compete for harmonic measure, and show how the tree structure is related to coexistence in this model.


Nous étudions la structure de l’agrégat limite $A_{\infty}=\bigcup_{n\geq0}A_{n}$ du DLA en dimension 1 avec longue portée, tel qu’introduit dens (Ann. Probab. 44 (2016) 3546–3579). Nous montrons (sous des hypothèses de régularité) que pour des marches aléatoires admettant un moment d’ordre 3, $A_{\infty}$ a une structure de renouvellement et une densité positive, tandis que pour les marches ayant seulement une variance finie, la structure de renouvellement disparaît et la densité est nulle. Nous définissions une structure arborescente sur l’agrégat et montrons quelques résultats sur les degrés et le nombre de bouts de ces arbres aléatoires. Nous introduisons un nouveau modèle de « compétition harmonique » entre des particules de couleurs différentes, et nous montrons que la structure d’arbre est reliée au problème de coexistence dans ce modèle.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1513-1527.

Received: 6 April 2015
Revised: 11 November 2015
Accepted: 15 November 2015
First available in Project Euclid: 27 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C24: Interface problems; diffusion-limited aggregation
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 97K50: Probability theory 97K60: Distributions and stochastic processes

Diffusion limited aggregation DLA Random walk Harmonic measure Phase transition Renewal structure


Amir, Gideon. One-dimensional long-range diffusion-limited aggregation III – The limit aggregate. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1513--1527. doi:10.1214/15-AIHP731.

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