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Small-Particle Limits in a Regularized Laplacian Random Growth Model
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2015 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 334, no 1, 331-366 p.Article in journal (Refereed) Published
Abstract [en]

We study a regularized version of Hastings-Levitov planar random growth that models clusters formed by the aggregation of diffusing particles. In this model, the growing clusters are defined in terms of iterated slit maps whose capacities are given by where c > 0 is the capacity of the first particle, {I broken vertical bar (n) } (n) are the composed conformal maps defining the clusters of the evolution, {theta (n) } (n) are independent uniform angles determining the positions at which particles are attached, and sigma > 0 is a regularization parameter which we take to depend on c. We prove that under an appropriate rescaling of time, in the limit as c -> 0, the clusters converge to growing disks with deterministic capacities, provided that sigma does not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow over longer time periods showing that, by letting alpha -> 0 at different rates, this flow converges to either the Brownian web on the circle, a stopped version of the Brownian web on the circle, or the identity map. As the harmonic measure flow is closely related to the internal branching structure within the cluster, the above three cases intuitively correspond to the number of infinite branches in the model being either 1, a random number whose distribution we obtain, or unbounded, in the limit as c -> 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis.

Place, publisher, year, edition, pages
2015. Vol. 334, no 1, 331-366 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-246821DOI: 10.1007/s00220-014-2158-yISI: 000348919300010OAI: oai:DiVA.org:uu-246821DiVA: diva2:795137
Available from: 2015-03-13 Created: 2015-03-10 Last updated: 2017-12-04Bibliographically approved

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Viklund, Fredrik Johansson

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