Results at zero temperature and the power-law
5.4 Clusters and the watermark
5.4.1 Basic concepts
UµS(x)−UµL(x) †
UµS(x)−UµL(x)
. (5.12)
The comparison of smearing and Laplace filtering via the link difference confirms the re-sults of optimal filtering. Figure 5.5 shows plots both for periodic and anti-periodic bound-ary conditions of the lattice Laplace operator. In both plots, we can see that the position of weak filtering (80 Laplace modes, 10 smearing steps) and strong filtering (20 Laplace modes, 20 smearing steps) lie in a “valley” where the difference of links is smallest.
5.4 Clusters and the watermark
A cluster analysis can be done for many observables that are sufficiently smooth. In this work, clusters of the topological charge density will be in the focus. I call the method to choose some kind of cutoff in the topological charge density the watermark method.
5.4.1 Basic concepts
I will first give the definition of a cluster using the “watermark” method. Choosing the a watermark means that I pick a fixed fraction of points f from every configuration in the ensemble. I pick those pointsxwhere the absolute value of the topological charge|q(x)|is largest. These are the points above the watermark. They are then grouped into clusters.
Two points belong to the same cluster if they are nearest neighbors, i.e., if the lattice distance is smaller than a fixed value, typically 1. Figure 5.6 shows an artist’s concept of the watermark. The “water” is lowered until a fixed fraction of points f is above the watermark.
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1 10 100
1 10 100 Difference of links,
periodic boundary conditions 0.24
0.2 0.16 0.14 0.12 0.1 0.08 0.07 0.06 0.055 0.05 0.045
Laplace modes
smearing steps
1 10 100
1 10 100 Difference of links,
anti-periodic boundary conditions 0.24
0.2 0.16 0.14 0.12 0.1 0.08 0.07 0.06 0.055 0.05 0.045
Laplace modes
smearing steps
Figure 5.5: This figure shows the difference of links. The contours show the difference of links computed via Eq. (5.12). The label “
•
” shows the position of weak filtering, the label “N” shows the position of strong filtering. The strong and weak filtering positions appear in the “valley” of least link differences. The link difference has been evaluated on one configuration.5.4 Clusters and the watermark 39 of 100
Figure 5.6: This figure shows an artist’s concept of the watermark. The gray bars represent lattice points and the height corresponds to the topological charge.
This method is more convenient than choosing a fixed cutoff, because the maximal topological charge density varies greatly among configurations. Had I chosen a fixed cutoff, I would have gotten – in the same ensemble – a lot of configurations with no clusters at all and at the same time many configurations with dozens of clusters. Quite understandably this has a negative effect on the signal to noise ratio. Figure 5.7 shows the relation between the watermark and the cutoff in the topological charge density. We can see that the watermark method corresponds to large steps in the cutoff if only a small fraction of points is taken into account. Later, the cutoff decreases only very slowly with the watermark. From the curve of the Dirac filtered configuration, we deduce that the same watermark translates into very different cutoffs, as we can see by the large error bars. Note that the Dirac filtered topological charge density has been calculated from just 10 configurations, but the Laplace filtered and the smeared topological charge density has been calculated from 295 configurations.
Before we finally turn to the analysis of the clusters, I want to present one more quantity to show the agreement of the various filtering methods. In the previous section, Section 5.3, “Optimal filtering,” we have convinced ourselves that the strong and weak filtering parameters are indeed optimal filtering parameters. One last consistency check now follows. Like clustering, it is also based on the concept of a watermark. For this check, I will introduce the relative point overlap. The idea is now to compare the topological charge densities directly. The topological charge density is computed via two different filtering methods A and B. The relative point overlap sAB between two methodsA and B at a fixed fraction of points is given by:
sAB = P
x∈XA∩XB
qA(x)qB(x)>0
1 P
x∈XA∪XB 1 . (5.13)
Here,XA,B are all points that are above the watermark for methodA, B. This just means
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0 0.2 0.4 0.6 0.8 1 1.2
1 10 100 1000 10000 100000
8 Dirac modes 20 Laplace modes 20 smearing steps
Figure 5.7: This figure shows how the watermark translates – on average – to a cutoff in the topological charge density. The abscissa shows the number of points that contribute to the cluster analysis. The ordinate shows the charge density that corresponds to that, normalized to the average maximum. For smeared and Laplace filtered configurations, the curves are an average over the full ensemble of 295 configurations, for the Dirac filtered topological charge density, the curve shows the average of 10 configurations.
that sAB is the number of points that are above the watermark for both methods in common and where the sign of the topological charge density agrees, divided by the number of points above the watermark for any method. Figure 5.8 shows plots for the relative point overlap at weak and strong filtering, as well as a pictogram how it is calculated. We can see that the agreement of all three methods is quite good for all values of the fraction of points f.