Stationary Harmonic Measure and DLA in the Upper half Plane

TL;DR

This paper introduces the stationary harmonic measure in the upper half plane and analyzes the growth bounds of diffusion limited aggregation (DLA) processes, providing new bounds on their maximum height over time.

Contribution

It defines the stationary harmonic measure in the upper half plane and establishes new upper bounds for the growth rate of DLA processes in both discrete and continuous time.

Findings

Continuous DLA growth rate is bounded by o(t^{2+ε})

Discrete DLA maximum height is bounded by o(n^{2/3+ε})

Develops an interface growth process to bound growth according to stationary harmonic measure

Abstract

In this paper, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(t^{2+\epsilon})$. When time is discrete, we also prove a better upper bound of $o(n^{2/3+\epsilon})$, on the maximum height of the aggregate at time $n$. An important tool developed in this paper, is an interface growth process, bounding any process growing according to the stationary harmonic measure. Together with [10] one obtains non zero growth rate for any such process.