Small particle limits in a regularized Laplacian random growth model

الأصل · 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 cₙ=c|Φₙ₋₁'(eσ⁺ⁱθⁿ)|⁻α, α≥ 0, where c>0 is the capacity of the first particle, Φₙₙ are the composed conformal maps defining the clusters of the evolution, θₙₙ 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 converges to 0, the clusters converge to growing disks with deterministic capacities, provided that σdoes not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow, showing that by letting αtend 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.