Behaviour of the harmonic measure at the bottom of fjords

Abstract

Diffusion-limited aggregates are among many important fractal shapes that involve deep indentations usually called fjords. To estimate the harmonic measure at the bottom of a fjord seems a prohibitive task, but the authors find that a new mathematical equality due to Beurling, Carleson and Jones makes it easy. They find that the harmonic measure at the bottom of a fjord, as a function of its Euclidean depth, can exhibit a wide range of behaviours. They introduce an infinite family of model fjords, for which the equality takes a very simple form. In this family the decay of the harmonic measure at their bottoms can be, for example, power law, semi-exponential, stretched exponential and exponentially stretched exponential. They show that self-affinity or randomness can lead to faster than power law decays of the minimal growth probability on boundaries.