On sets of zero stationary harmonic measure

TL;DR

This paper characterizes the conditions under which subsets have zero or non-zero stationary harmonic measure based on their horizontal growth, providing foundational insights for stationary DLA studies and related aggregation processes.

Contribution

It establishes a clear relationship between horizontal growth rates of subsets and their stationary harmonic measure, a key step for future research on stationary DLA.

Findings

Subsets with sub-linear horizontal growth have non-zero stationary harmonic measure.

Subsets with super-linear horizontal growth have zero stationary harmonic measure.

Growth processes proportional to stationary harmonic measure maintain non-zero measure over time.

Abstract

In this paper, we prove that any subset with an appropriate sub-linear horizontal growth has a non-zero stationary harmonic measure. On the other hand, we also show any subset with super-linear horizontal growth will have a $0$ stationary harmonic measure at every point. This result is fundamental to any future study of stationary DLA. As an application we prove that any possible aggregation process with growth rates proportional to the stationary harmonic measure has non zero measure at all times.