Scaling limits of anisotropic growth on logarithmic time-scales

TL;DR

This paper investigates the long-term behavior of anisotropic growth models on logarithmic time-scales, revealing a stochastic transition from unstable to stable states driven by Gaussian fluctuations.

Contribution

It introduces a detailed analysis of the harmonic measure evolution over logarithmic time-scales, uncovering a stochastic transition characterized by Gaussian variables.

Findings

Harmonic measure flow becomes stochastic over logarithmic time-scales.

Existence of a critical time window for stochastic transition.

Trajectory characterized by a single Gaussian random variable.

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.