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Small-particle scaling limits in a regularized Laplacian growth model

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We study a regularized version of the Hastings-Levitov model of Laplacian random growth. In addition to the usual feedback parameter alpha>0, this regularized version features a smoothing parameter sigma>0. Simulations of the resulting growth processes reveal non-trivial features that differ from those observed in HL(0). We prove convergence of random clusters, in the limit as the size of the individual aggregating particles tends to zero, to deterministic limits, provided the smoothing parameter does not tend to zero too fast. We also study scaling limits of the harmonic measure flow on the boundary, and show that it can be described in terms of stopped Brownian webs on the circle. In contrast to the case alpha=0, the flow does not always collapse into a single Brownian motion, which can be interpreted as a random number of infinite branches being present in the clusters. Joint work with Fredrik Johansson Viklund and Amanda Turner.

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