\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 mass 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 literature, $1.71 \pm 0.01$ 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 mass method 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 Errors
Taking an average of the trailing $5000$ readings we come to a value of $fd = 1.73$. As can be seen on the figure this is divergence from the literature (we suspect due to the gridded nature of the embedding space) the result is reasonable and consistent across runs. We consider this, along with the sourcing of the initially provided code, to be sufficient grounding the start of our trust chain.

This also allows us to say with reasonable confidence that we can halt our model around $N_C = 5000$ as a 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}

\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 data was obtained in two batches: in the $p_{stick} \in [0.1, 1]$ range $100$ samples were taken with different seeds with $N_C= 2000$, the fractal dimension being computed by the mass method; for the $p_{stick} \in (0.001, 0.1)$ range a $100$ samples of $N_C = 5000$ clusters were used.
}
\label{sp-fd}
\end{figure}

The first alteration we shall make to the DLA model is the introduction of a probabilistic component to the sticking behaviour. We parametrise this behaviour by a sticking probability $p_{stick} \in (0, 1]$, 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 first the clusters for different values of $p_{stick}$ we can see in Figure \ref{sp-dla-comparison} a clear thickening of the arms with lower values of $p_{stick}$. This aligns with data for the fractal dimension, as seen in Figure \ref{sp-fd}, with thicker arms bringing the cluster closer to a non-fractal two dimensional object.

In the low $p_{stick}$ domain we record values of $\mathrm{fd} > 2$ which is the embedding domain. This is unexpected and points towards a possible failure of our mass-method fractal dimension calculation. More work and analysis is required to verify these results.
% TODO Conclusion point

As discussed in the Appendix, \nameref{generic-dla}, this also provides the next chain of grounding between the initially provided code, and the new generic framework. Further details can be found in the aforementioned appendix.

\subsection*{Higher Dimensions}

The next alteration to explore is changing the embedding space to be higher dimensional. Here we use a k-dimensional tree structure to store the aggregate as opposed to an array based grid allowing us to greatly reduce memory consumption ($O(\text{grid\_size}^D) \to O(n)$ where $n$ is the number of particles in the aggregate) whilst retaining a strong access and search time complexity of $O(n \log n)$\cite{bentleyMultidimensionalBinarySearch1975}.

To start we model two styles of random walk: direct, where only those cells which are directly adjacent to its current location current location are accessible; off-axis, where all the full $3 \times 3 \times 3$ cubic (bar the centre position) are available. The ($N_c, fd$) correspondence is shown in Figure \ref{3d-nc-fd-convergence} where we can see that both walk methods, as expected produce identical results, varying only slightly from a naive implementation in the initially provided codebase included to ensure correct behaviour. These off axis walks do however offer a speed boost as the larger range of motion leads to faster movement within the space.

Modelling the system across the range of $p_{stick}$ we obtain results as shown in Figure \ref{sp-fd-2d-3d}. These show a similar pattern as was seen in the 2D case of Figure \ref{sp-fd}. We note that whilst these lines are similar, they are not parallel showing distinct behaviour.

At this moment we do not have an analytical form for this relation but further work may provide such.
% TODO Conclusion point

\begin{figure}
\includegraphics[width=\columnwidth]{figures/sp-fd-2d-3d}
\caption{A comparison of the fractal dimension of DLA aggregates in 2- and 3-dimensional embedding space. The datasets were obtained by averages of $100$ and $200$ for 2D and 3D respectively, both with data recorded at a increment of $\Delta p_{stick} \approx 0.1$, and an aggregate size of $2000$.}
\label{sp-fd-2d-3d}
\end{figure}

\begin{figure}
\includegraphics[width=\columnwidth]{figures/3d-nc-fd-convergence}
\caption{A comparison of direct and off-axis walks in 3 dimensions, using both the new framework (NF)  and the initial provided code (IPC). Note a slight divergence between the NF and IPC lines but a complete agreement between the direct and off-axis walks for the NF. Errors are not displayed as they are to small to be visible on this graph due to the large sample size.} % TODO Verify literature line here and rename it.
\label{3d-nc-fd-convergence}
\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.

% MORE EXTENSIONS

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

\section*{Conclusion}

In conclusion, we have validated that the new framework provides consistent behaviour aligned with previous models, whilst allowing for a wider range of behaviours to be easily tested. Future work is required to determine analytic or physical explanations for the data presented. However there are a number of pointers to places of immediate interest and avenues waiting to be explored.

