FIG. 4: Correlation length scales. The cross cor- relation between δT$_{p}$ and Δ as a function of sepa- ration, and the autocorrelations of Δ and T$_{p}$ , radi- ally averaged. The cross correlation shows that areas with large T$_{p}$ values for their gap size are on average surrounded by larger gap regions. The autocorrela- tions show that T$_{p}$ has stronger correlation on length scales above 1 nm. The results are not significantly different if averaged along the a or b axis instead of radially. to depend only locally on the gap, the distribu- tion of T$_{p}$ values with the same gap would only be due to random fluctuations. Therefore, when we subtract the mean T$_{p}$ value (based on gap size) from each measured T$_{p}$ value, the result should be uncorrelated noise. However, figure 3b shows a significant spatial pattern associated with this quantity, which we call δT$_{p}$ . We can understand the meaning of δT$_{p}$ and relate it to the gap map using the cross corre- lation as a function of separation distance. We define the cross correlation as where σ$_{δT}$$_{p}$ and σ$_{Δ}$ are the standard deviations of δT$_{p}$ and Δ respectively, and y runs over the N available data points. Although the supermodu- lation on the b axis does have a small correlation with the gap, in practice ρ is isotropic within the noise, so we plot a circular average in figure 4. For zero separation, the cross correlations math- ematically must be zero, because we have ex- plicitly eliminated the local relation between Δ and T$_{p}$ when we subtract the mean to produce δT$_{p}$ . The peak at positive separation indicates that T$_{p}$ at a particular point is significantly de- termined by gaps 1 nm away, which we attribute to the proximity effect. In addition, there is a long tail of positive correlation, indicating that T$_{p}$ is also dependent on gaps several nanome- ters away. Figure 4 also shows the autocorrela- tion of Δ and T$_{p}$ averaged along both axes and plotted as a function of separation, showing that the pairing temperature has a longer length scale (1.1 nm at half-maximum) than the gap (0.9 nm at half-maximum). These correlations occur on length scales longer enough than the 0.25 nm pixel spacing to rule out “smoothing” due to the interpolation used in aligning between different temperatures. The strength of T$_{p}$ correlations is also higher than that of gap correlations over the tail of the distribution (1 – 6 nm). Although 6 ρ ( x ) = 1 Nσ$_{δT}$$_{p}$σ$_{Δ}$ ∑ y δT$_{p}$ ( y )Δ( y − x ) , (1) Distance (nm) Correlation 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 T$_{p}$  T$_{p}$   T$_{p}$