Scaling limits of anisotropic Hastings–Levitov clusters

Fredrik Johansson Viklund; Alan Sola; Amanda Turner

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 1, page 235-257
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Johansson Viklund, Fredrik, Sola, Alan, and Turner, Amanda. "Scaling limits of anisotropic Hastings–Levitov clusters." Annales de l'I.H.P. Probabilités et statistiques 48.1 (2012): 235-257. <http://eudml.org/doc/271973>.

@article{JohanssonViklund2012,
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.},
author = {Johansson Viklund, Fredrik, Sola, Alan, Turner, Amanda},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {anisotropic growth models; scaling limits; Loewner differential equation; boundary flow},
language = {eng},
number = {1},
pages = {235-257},
publisher = {Gauthier-Villars},
title = {Scaling limits of anisotropic Hastings–Levitov clusters},
url = {http://eudml.org/doc/271973},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Johansson Viklund, Fredrik
AU - Sola, Alan
AU - Turner, Amanda
TI - Scaling limits of anisotropic Hastings–Levitov clusters
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 1
SP - 235
EP - 257
AB - 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.
LA - eng
KW - anisotropic growth models; scaling limits; Loewner differential equation; boundary flow
UR - http://eudml.org/doc/271973
ER -

References

top
  1. [1] R. A. Arratia. Coalescing Brownian motions on the line. Ph.D. thesis, Univ. Wisconsin, 1979. MR2630231
  2. [2] R. O. Bauer. Discrete Löwner evolution. Ann. Fac. Sci. Toulouse Math. (6) 12 (2003) 433–451. Zbl1054.60102MR2060594
  3. [3] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1999. Zbl0172.21201MR1700749
  4. [4] M. Björklund. Ergodic theorems for random clusters. Stochastic Process. Appl.120 (2010) 296–305. Zbl1191.60065MR2584895
  5. [5] L. Carleson and N. Makarov. Aggregation in the plane and Loewner’s equation. Comm. Math. Phys.216 (2001) 583–607. Zbl1042.82039MR1815718
  6. [6] L. Carleson and N. Makarov. Laplacian path models. Dedicated to the memory of Thomas H. Wolff. J. Anal. Math.87 (2002) 103–150. Zbl1040.30011MR1945279
  7. [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. E87 (1999) 1366–1378. MR1672801
  8. [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. Zbl0104.13801MR136460
  9. [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. Zbl1105.60075MR2094432
  10. [10] J. B. Garnett. Bounded Analytic Functions, reviewed 1st edition. Graduate Texts in Mathematics 236. Springer, New York, 2007. Zbl0469.30024MR2261424
  11. [11] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, 6th edition. Academic Press, San Diego, 2000. Zbl0981.65001MR1773820
  12. [12] M. Hastings and L. Levitov. Laplacian growth as one-dimensional turbulence. Phys. D116 (1998) 244–252. Zbl0962.76542
  13. [13] F. Johansson and A. Sola. Rescaled Lévy–Loewner hulls and random growth. Bull. Sci. Math.133 (2009) 238–256. Zbl1167.30007MR2512828
  14. [14] R. Julien, M. Kolb and R. Botet. Diffusion limited aggregation with directed and anisotropic diffusion. J. Physique45 (1984) 395–399. 
  15. [15] O. Kallenberg. Random Measures, 3rd edition. Akademie-Verlag, Berlin, 1983. Zbl0544.60053MR818219
  16. [16] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, New York, 2002. Zbl0892.60001MR1876169
  17. [17] G. Lawler. Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI, 2005. Zbl1074.60002MR2129588
  18. [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. Zbl1126.82011MR2044671
  19. [19] R. Malaquias, S. Rohde, V. Sessak and M. Zinsmeister. On Laplacian growth. To appear. 
  20. [20] J. Norris and A. Turner. Planar aggregation and the coalescing Brownian flow. Available at http://arxiv.org/abs/0810.0211. Zbl1327.60086
  21. [21] Ch. Pommerenke. Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften 299. Springer, Berlin–Heidelberg, 1992. Zbl0762.30001MR1217706
  22. [22] M. N. Popescu, H. G. E. Hentschel and F. Family. Anisotropic diffusion-limited aggregation. Phys. Rev. E 69 (2004) 061403. 
  23. [23] S. Rohde. Personal communication, 2008. 
  24. [24] S. Rohde and M. Zinsmeister. Some remarks on Laplacian growth. Topology Appl.152 (2005) 26–43. Zbl1077.60040MR2160804
  25. [25] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge, 1999. Zbl0973.60001
  26. [26] B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields111 (1998) 375–452. Zbl0912.60056MR1640799
  27. [27] T. A. Witten, Jr. and L. M. Sander. Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett.47 (1981) 1400–1403. MR704464

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.