
\section*{Results}

\subsection*{Preliminary Work: Testing Initial Implementation and Fractal Dimension Calculations}
\label{ii-fdc}

\begin{figure}
\includegraphics[width=\columnwidth]{figures/rmax-n.png}
\caption{The growth of $N$ vs $r_{\mathrm{max}}$ for $20$ runs of the standard DLA model. When fitting the first $1000$ points are not included when fitting to improve accuracy. The 
 $50$ data points are not included as the data contains to much noise to be meaningfully displayed. Also included in the figure is the value from theory, $1.71 \pm 0.01$ from \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
\label{rmax-n}
\end{figure}

\begin{figure}[t!]
\includegraphics[width=\columnwidth]{figures/nc-fd-convergence.png}
\caption{The converge of the fractal dimension of $20$ runs of the standard DLA model. This uses the the simple calculation method. The first $50$ data points are not included as the data contains to much noise to be meaningfully displayed. Also included in the figure is the value from theory, $1.71 \pm 0.01$ from \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
\label{nc-fd-convergence}
\end{figure}

To start we do $20$ runs, with seeds $1, 2, \dots, 20$, of the standard DLA model using the minimally altered initially provided code. The fractal dimension is calculated using the simple method \todo{do I want to ref this?} and averaged across the $20$ runs. This is shown in Figure \ref{nc-fd-convergence} along with the result from literature, $d = 1.7 \pm 0.6$ \cite[Table 1, $\langle D(d = 2)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.
\todo{Do I need to find a grid value for this I think this might be continuous}

Taking an average of the trailing $5000$ readings we come to a value of $d = 1.73$ \todo{errors and rounding}, while this values diverges slightly from the value reported in literature, this result provides a reasonable grounding for our model as being roughly correct and provides a useful point of comparison for future work. 

This also allows us to say with reasonable confidence that we can halt our model around $N_C = 5000$ as a reasonable trade off between computational time and accuracy. This should be verified for particular model variations however.

\subsection{Probabilistic Sticking}

\begin{figure}[t!]
\includegraphics[width=\columnwidth]{figures/sp-fd}
\caption{The fractal dimension for the DLA system on a 2D grid lattice a sticking probability $p_{stick}$. This value was obtained from $100$ runs with different seeds, by computing the value of the fractal dimension using the simple method, taking a mean across the last $100$ measurements on a $2000$ particle cluster.
% TODO These numbers are way too small given the results of Figure 1.
}
\label{sp-fd}
\end{figure}

As discussed one of the possible alterations of the system is the introduction of a probabilistic component to the sticking behaviour of the DLA system. Here we introduced a probability $p_{stick}$ to the initial grid based sticking behaviour of the particles, with the particle being given this probability to stick at each site (for example, if the particle was adjacent to two cells in the aggregate, then the probabilistic aspect would apply twice).

This was also the case used to ground both the minimally altered code, and our new generic system, to ensure they are functioning correctly. The data for both is presented in Figure \ref{sp-fig}. As we would expect we see a g

\subsection*{Higher Dimensions}

\subsection*{Hexagonal}
