\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 to a maximum value of $N_C = 10000$. Also included is a line of best fit for the data, less the first $50$ which are removed to improve accuracy, with form $\log N_C = a_0 + \mathrm{fd} \cdot \log r_{\mathrm{max}}$ and coefficients $\mathrm{fd} = 1.7685 \pm 0.0004$, $a_0 = -0.1815 \pm 0.002$. % TODO Verify rounding
}
\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, $\mathrm{fd} = 1.71 \pm 0.01$ 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 IPC. We use both the instantaneous and	 line-fitting mass method, as shown in Figure \ref{nc-fd-convergence} and Figure \ref{rmax-n} respectively. For the instantaneous case, 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, $\mathrm{fd} = 1.7 \pm 0.6$ \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 $\mathrm{fd} = 1.735 \pm 0.020$. 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 IPC, 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. However care must be taken to verify this is appropriate for any particular model variation.

\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 $\mathrm{fd} > 2$, greater that the 2D space they are embedded within. 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.

%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 (see the aforementioned appendix for more).

\subsection*{Higher Dimensions}

\begin{figure}[hbt]
\includegraphics[width=\columnwidth]{figures/3d-eg}
\caption{A 3D DLA aggregate on a 3D orthogonal grid, with $N_C = 5000$, coloured by deposition time. Note that the view of the particles as spheres is an artifact of the rendering process, they are in fact cubes. Here we can observe the expected tendril structure of a DLA aggregate.}
\label{3d-eg}
\end{figure}

\begin{figure}[hbt]
\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}[hbt]
%\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. Also included is the result from literature, $\mathrm{fd} = 2.51 \pm 0.01$\cite[Table 1, $\langle D(d = 3)\rangle$]{nicolas-carlockUniversalDimensionalityFunction2019}.}
%\label{3d-nc-fd-convergence}
%\end{figure}

The next alteration to explore is changing the embedding space to be higher dimensional, an example of which can be seen in Figure \ref{3d-eg}. 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 forms of random walk: direct, where the particle can only access directly adjacent cells and off-axis, where all the full $3 \times 3 \times 3$ cubic (bar the centre position) are available. These behave identically to each other, varying only slightly from a naive implementation in the IPC included to provide assurance of the 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.

Note that the divergence from the value expected from the literature is greater here than in the 2D case, with a fractal dimension reported of $\mathrm{fd} = 2.03 \pm 0.06$. This along with our inability to find a satisfactory analytic form for this behaviour suggests further analysis on different grids is required.

% Extensions Do I want to do higher dimensions still 4d?
% 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.

\section*{Conclusion}

In this report we have presented findings for the fractal dimension of DLA aggregates in 2D and 3D on orthogonal grids, as well as qualitative assessments of the variation of the fractal dimension across a range of sticking probabilities. In addition, we have validated the framework used to be consistent with previous models allowing for quick iteration. Future work is required to determine analytic or physical explanations for the data presented, specifically the $(p_{stick}, \mathrm{fd})$ relation, in addition to identifying the cause of the divergence between reported results and previous models, with literature, possibly through different geometries.
