Scaling limits of anisotropic growth on logarithmic time-scales

George Liddle; Amanda Turner

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arXiv · 2211.03676

Scaling limits of anisotropic growth on logarithmic time-scales

George Liddle·Amanda Turner

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Abstract

We study the anisotropic version of the Hastings-Levitov model AHL$(\nu)$. Previous results have shown that on bounded time-scales the harmonic measure on the boundary of the cluster converges, in the small-particle limit, to the solution of a deterministic ordinary differential equation. We consider the evolution of the harmonic measure on time-scales which grow logarithmically as the particle size converges to zero and show that, over this time-scale, the leading order behaviour of the harmonic measure becomes random. Specifically, we show that there exists a critical logarithmic time window in which the harmonic measure flow, started from the unstable fixed point, moves stochastically from the unstable point towards a stable fixed point, and we show that the full trajectory can be characterised in terms of a single Gaussian random variable.

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George Liddle, Amanda Turner. 2022-11-07. Scaling limits of anisotropic growth on logarithmic time-scales. https://arxiv.org/abs/2211.03676

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