Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of anisotropic Hastings–Levitov clusters

Fredrik Johansson Viklund, Alan Sola, and Amanda Turner

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We consider a variation of the standard Hastings–Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations around the deterministic limit flow.


Dans cet article, on presente une étude d’une version du modèle de Hastings–Levitov HL (0) où la croissance est anisotrope. Deux limites d’échelle naturelles sont établies, et nous décrivons précisément les effets de l’anisotropie. Nous montrons que les formes limites du modèle peuvent être réalisées comme remplissages associés à l’équation de Loewner et que l’évolution de la mesure harmonique sur la frontière des agrégats tend vers un certain flot deterministe. Nous caractérisons enfin les fluctuations stochastiques autour de ce flot.

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Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 1 (2012), 235-257.

First available in Project Euclid: 23 January 2012

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Primary: 30C35: General theory of conformal mappings 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F99: None of the above, but in this section

Anisotropic growth models Scaling limits Loewner differential equation Boundary flow


Johansson Viklund, Fredrik; Sola, Alan; Turner, Amanda. Scaling limits of anisotropic Hastings–Levitov clusters. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 1, 235--257. doi:10.1214/10-AIHP395.

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