\section*{Results}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{figures/rmax-n.png}
\caption{The growth of $N$ vs $r_{\mathrm{max}}$ for $20$ runs of the standard DLA model. Also included is a line of best fit for the data, less the first $50$ which are removed to improve accuracy.
% TODO Check all of my captions are correct.
% TODO Add information for this
}
\label{rmax-n}
\end{figure}

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

\begin{figure}[hbt]
\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}[hbt]
  \includegraphics[width=\columnwidth]{figures/eg-across-sp/sp-range.png}
  \caption{Here we see the result of three different DLA simulations with $p_{stick} = 0.1,0.5,1.0$ from left to right. Note the thickening of the arms at low probabilities.}
\label{sp-dla-comparison}
\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).

Comparing the results from different runs we can see in Figure \ref{sp-dla-comparison} the clear thickening of the arms with lower values of $p_{stick}$. This aligns with your observation of the fractal dimension, as seen in Figure \ref{sp-fd}.

\begin{figure}[hbt]
\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}

This was also the case used to ground both the minimally altered code, and our new generic system, to ensure they are functioning correctly.  This is discussed in more depth in the Appendix, \nameref{generic-dla}.

\subsection*{Higher Dimensions}

The next alteration to explore is changing the embedding space to be higher dimensional. This is an excellent example of the versatility of the generic DLA framework as for higher dimensions it becomes advantageous to move from an array based grid storage to a k-dimensional tree structure for more efficient storage, while only moving to $O(\log n)$ average performance for searches and inserts\cite{bentleyMultidimensionalBinarySearch1975}.

Running with various values of $p_{stick}$ we get the results shown in Figure \ref{sp-fd-2d-3d}.


\begin{figure}
\includegraphics[width=\columnwidth]{figures/3d-nc-fd-convergence}
\caption{TODO}
\label{3d-nc-fd-convergence}
\end{figure}

\begin{figure}
\includegraphics[width=\columnwidth]{figures/sp-fd-2d-3d}
\caption{TODO}
\label{sp-fd-2d-3d}
\end{figure}

% TODO Do I want to do higher dimensions still 4d?
% TODO how am I going to cope with the fact we don't agree with theory?
% TOOD Look at theory to see if I can find a curve for these sp-fd graphs or at the very least note similarities and differences between them. "Given the erroneous behaviour for low sp we are uncertain as to the correctness). Maybe take another crack at boxcount since you've mentioned it and it might be interesting.

\begin{enumerate}
	\item The next obvious extension is 3D
	\item Try with on axis and off-axis movement
	\item See that off-axis has no effect but is quicker as traverses space quicker (I mean also validate this)
	\item Now try 3D + SP
\end{enumerate}

% MORE EXTENSIONS

\subsection*{Continuous Space}

\begin{enumerate}
	\item We get a divergence from theory, what happens if we use continuous
\end{enumerate}

\subsection*{Hexagonal}

\subsection*{External force onto wall}
