We study the anisotropic version of the Hastings-Levitov model AHL. 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.
Supported by EPSRC grant EP/T027940/1.
"Scaling limits of anisotropic growth on logarithmic time-scales." Electron. J. Probab. 28 1 - 32, 2023. https://doi.org/10.1214/23-EJP964