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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Small-particle limits in a regularized Laplacian random growth model
AU - Johansson Viklund, Fredrik
AU - Sola, Alan
AU - Turner, Amanda
N1 - 35 pages, 20 figures
PY - 2015/2/1
Y1 - 2015/2/1
N2 - 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|\Phi_{n-1}'(e^{\sigma+i\theta_n})|^{-\alpha}, \alpha \geq 0, where c>0 is the capacity of the first particle, {\Phi_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 converges to 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, showing that by letting \alpha tend to 0 at different rates it 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 converges to 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis.
AB - 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|\Phi_{n-1}'(e^{\sigma+i\theta_n})|^{-\alpha}, \alpha \geq 0, where c>0 is the capacity of the first particle, {\Phi_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 converges to 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, showing that by letting \alpha tend to 0 at different rates it 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 converges to 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis.
KW - Random growth models
KW - Scaling limits
KW - Loewner differential equation
KW - Harmonic measure flow
KW - Brownian web
U2 - 10.1007/s00220-014-2158-y
DO - 10.1007/s00220-014-2158-y
M3 - Journal article
VL - 334
SP - 331
EP - 366
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 1
ER -