Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of anisotropic Hastings–Levitov clusters

Fredrik Johansson Viklund, Alan Sola, and Amanda Turner

Full-text: Open access

Abstract

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.

Résumé

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.

Article information

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

Dates
First available in Project Euclid: 23 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1327328021

Digital Object Identifier
doi:10.1214/10-AIHP395

Mathematical Reviews number (MathSciNet)
MR2919205

Zentralblatt MATH identifier
1251.82025

Subjects
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

Keywords
Anisotropic growth models Scaling limits Loewner differential equation Boundary flow

Citation

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. https://projecteuclid.org/euclid.aihp/1327328021


Export citation

References

  • [1] R. A. Arratia. Coalescing Brownian motions on the line. Ph.D. thesis, Univ. Wisconsin, 1979.
  • [2] R. O. Bauer. Discrete Löwner evolution. Ann. Fac. Sci. Toulouse Math. (6) 12 (2003) 433–451.
  • [3] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1999.
  • [4] M. Björklund. Ergodic theorems for random clusters. Stochastic Process. Appl. 120 (2010) 296–305.
  • [5] L. Carleson and N. Makarov. Aggregation in the plane and Loewner’s equation. Comm. Math. Phys. 216 (2001) 583–607.
  • [6] L. Carleson and N. Makarov. Laplacian path models. Dedicated to the memory of Thomas H. Wolff. J. Anal. Math. 87 (2002) 103–150.
  • [7] B. Davidovitch, H. G. E. Hentschel, Z. Olami, I. Procaccia, L. M. Sander and E. Somfai. Diffusion limited aggregation and iterated conformal maps. Phys. Rev. E 87 (1999) 1366–1378.
  • [8] M. Eden. A two-dimensional growth process. In Proc. 4th Berkeley Sympos. Math. Statist. and Probab., Vol. IV 223–239. Univ. California Press, Berkeley, CA, 1961.
  • [9] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web: Characterization and convergence. Ann. Probab. 32 (2004) 2857–2883.
  • [10] J. B. Garnett. Bounded Analytic Functions, reviewed 1st edition. Graduate Texts in Mathematics 236. Springer, New York, 2007.
  • [11] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, 6th edition. Academic Press, San Diego, 2000.
  • [12] M. Hastings and L. Levitov. Laplacian growth as one-dimensional turbulence. Phys. D 116 (1998) 244–252.
  • [13] F. Johansson and A. Sola. Rescaled Lévy–Loewner hulls and random growth. Bull. Sci. Math. 133 (2009) 238–256.
  • [14] R. Julien, M. Kolb and R. Botet. Diffusion limited aggregation with directed and anisotropic diffusion. J. Physique 45 (1984) 395–399.
  • [15] O. Kallenberg. Random Measures, 3rd edition. Akademie-Verlag, Berlin, 1983.
  • [16] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, New York, 2002.
  • [17] G. Lawler. Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI, 2005.
  • [18] G. F. Lawler, O. Schramm and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004) 939–995.
  • [19] R. Malaquias, S. Rohde, V. Sessak and M. Zinsmeister. On Laplacian growth. To appear.
  • [20] J. Norris and A. Turner. Planar aggregation and the coalescing Brownian flow. Available at http://arxiv.org/abs/0810.0211.
  • [21] Ch. Pommerenke. Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften 299. Springer, Berlin–Heidelberg, 1992.
  • [22] M. N. Popescu, H. G. E. Hentschel and F. Family. Anisotropic diffusion-limited aggregation. Phys. Rev. E 69 (2004) 061403.
  • [23] S. Rohde. Personal communication, 2008.
  • [24] S. Rohde and M. Zinsmeister. Some remarks on Laplacian growth. Topology Appl. 152 (2005) 26–43.
  • [25] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge, 1999.
  • [26] B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111 (1998) 375–452.
  • [27] T. A. Witten, Jr. and L. M. Sander. Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47 (1981) 1400–1403.