Scaling limits of anisotropic growth on logarithmic time-scales

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