Small-Particle Limits in a Regularized Laplacian Random Growth Model

• Published in: Communications in mathematical physics 2015-02, Vol.334 (1), p.331-366
• Main Authors: Johansson Viklund, Fredrik, Sola, Alan, Turner, Amanda
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916; 1432-0916
• DOI: 10.1007/s00220-014-2158-y

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 c n = c | Φ n - 1 ′ ( e σ + i θ n ) | - α , α ≥ 0 , where c > 0 is the capacity of the first particle, {Φ n } n are the composed conformal maps defining the clusters of the evolution, { θ n } n are independent uniform angles determining the positions at which particles are attached, and σ > 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 σ 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 α → 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.