This repository contains my diploma thesis —Latex code and the generated pdf file.
In my thesis I consider internal diffusion limited aggregation—a random growth model—on
the two-dimensional lattice.
With each step, a new particle starts in the origin and performs a random walk
until it hits an unoccupied lattice point, where it stops.
I am interested in the asymptotic growth of the occupied cluster.
Lawler, Bramson, and Griffeath (Internal Diffusion Limited Aggregation, 1992) proved
that the asymptotic shape is a Euclidean ball.
This statement was remarkably improved by Jerison, Levine, and Sheffield (Logarithmic fluctuations for internal DLA, 2012),
who showed that the fluctuations from circularity are of logarithmic order.
This result is subject of my thesis.
I give a thorough overview of the proof by filling in the omitted details. One of
the steps that was left out in the paper and is filled in
in this thesis is an extension of the classical result that a harmonic
function of Brownian motion is a martingale to grid-harmonic functions
and grid Brownian motions. The proof of this statement constitutes the core of my thesis.
For further reading I refer to the thesis itself. Do not hesitate to write if you noticed a mistake (via https://www.researchgate.net/profile/Lennart_Clausen).