Abstract
We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Carathéodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ball R.C., Brady R.M., Rossi G., Thompson B.R.: Anisotropy and cluster growth by diffusion-limited aggregation. Phys. Rev. Lett. 55, 1406–1409 (1985)
Bazant, M.Z., Crowdy, D.: Conformal mapping methods for interfacial dynamics. In: Yip, S. ed., Handbook of Materials Modeling, Berlin-Heidleberg-NewYork:Springer, 2005, pp. 1417–1451
Carleson L., Makarov N.: Aggregation in the plane and Loewner’s equation. Commun. Math. Phys. 216(3), 583–607 (2001)
Carleson, L., Makarov, N.: Laplacian path models. J. Anal. Math. 87, 103–150 (2002). (Dedicated to the memory of Thomas H. Wolff)
Davidovitch B., Hentschel H.G.E., Olami Z., Procaccia I., Sander L.M., Somfai E.: Diffusion limited aggregation and iterated conformal maps. Phys. Rev. E (3) 59(2, part A), 1368–1378 (1999)
Eden, M.: A two-dimensional growth process. In: Proc. 4 th Berkeley Sympos. Math. Statist. and Prob. Vol. IV, Berkeley, CA: Univ. California Press, 1961, pp. 223–239
Fontes L.R.G., Isopi M., Newman C.M., Ravishankar K.: The Brownian web: characterization and convergence. Ann. Probab. 32(4), 2857–2883 (2004)
Hastings M.B.: Fractal to nonfractal phase transition in the dielectric breakdown model. Phys. Rev. Lett. 87, 175502 (2001)
Hastings M.B., Levitov L.S.: Laplacian growth as one-dimensional turbulence. Physica D 116(1–2), 244 (1998)
Jensen M.H., Levermann A., Mathiesen J., Procaccia I.: Multifractal structure of the harmonic measure of diffusion-limited aggregates. Phys. Rev. E 65, 046109 (2002)
Viklund F.J., Sola A., Turner A.: Scaling limits of anisotropic Hastings-Levitov clusters. Ann. Inst. H. Poincaré Probab. Stat. 48, 235–257 (2012)
Kesten H.: Hitting probabilities of random walks on Z d. Stoch. Proc. Appl. 25(2), 165–184 (1987)
Meakin P., Ball R.C., Ramanlal P., Sander L.M.: Structure of large two-dimensional square-lattice diffusion-limited aggregates: Approach to asymptotic behavior. Phys. Rev. A 35, 5233–5239 (1987)
Niemeyer L., Pietronero L., Wiesmann H.J.: Fractal dimension of dielectric breakdown. Phys. Rev. Lett. 57(5), 650 (1986)
Norris, J., Turner, A.G.: Planar aggregation and the coalescing Brownian flow. http://arxiv.org/abs/0810.0211v1 [math.PR], 2008
Norris, J., Turner, A.G.: Weak convergence of the localized disturbance flow to the coalescing Brownian flow. http://arxiv.org/abs/1106.3252v3 [math.PR], 2012
Rohde S., Zinsmeister M.: Some remarks on Laplacian growth. Topology Appl. 152(1-2), 26–43 (2005)
Witten T.A., Sander L.M.: Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47(19), 1400–1403 (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
James Norris: Research supported by EPSRC grant EP/103372X/1.
Rights and permissions
About this article
Cite this article
Norris, J., Turner, A. Hastings–Levitov Aggregation in the Small-Particle Limit. Commun. Math. Phys. 316, 809–841 (2012). https://doi.org/10.1007/s00220-012-1552-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1552-6