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Correlation

This was a somewhat complicated part of this whole process. I found that I could quite easily compute the correlation between the lattice elements on a single row using the Fast Fourrier Transform. This process extended quite simply to two-dimensional correlation using two-dimensional Fast Fourier Transform. Figures 35 and 36 demonstrates this correlation computation (on a lattice of size 256 by 256 after 128 iterations at temperature $T = 2.269$). In Figure 35, the height of the mesh at position $(i, j)$ is the amount of correlation between the spin associated with a place in the lattice and the place that is "over" by $i$ and "down" by $j$ (modulo the size of the lattice). After producing Figure 35, I recentered the spike to the center of the mesh and plotted a region near that spike. This is Figure 36.

Figure 35: Correlation mesh
\begin{figure}\begin{center} \mbox{\epsfig{figure=corr1.eps}} \end{center} \end{figure}

Figure 36: Detail of correlation mesh from Figure 35 after recentering.
\begin{figure}\begin{center} \mbox{\epsfig{figure=corr2.eps}} \end{center} \end{figure}


2005-05-05