4 6 8 10 12 log n $^{GLYPH<Minus>}$30 $^{GLYPH<Minus>}$25 $^{GLYPH<Minus>}$20 $^{GLYPH<Minus>}$15 $^{GLYPH<Minus>}$10 z $^{GLYPH<Minus>}$40 $^{GLYPH<Minus>}$30 $^{GLYPH<Minus>}$20 $^{GLYPH<Minus>}$10 z 0.2 0.4 0.6 0.8 y FIG. 12: y = n ( p − 1) vs K (left) and z = n ( q − 1) vs K (right) for the L × L -lattice, L = 6, 8, 10, 12, 14, 16, 32, 64, 128, 256, 512. The points represent K ∗ and the red line is at K$_{c}$ . The larger lattices have their points farther to the right in the plots. 40 0.1 0.2 0.3 0.4 0.5 K $^{GLYPH<Minus>}$0.4 $^{GLYPH<Minus>}$0.2 0.0 0.2 0.4 0.6 0.8 y 0.1 0.2 0.3 0.4 0.5 K $^{GLYPH<Minus>}$50 $^{GLYPH<Minus>}$40 $^{GLYPH<Minus>}$30 $^{GLYPH<Minus>}$20 $^{GLYPH<Minus>}$10 0 z The right plot of figure 11 shows y vs z for a range of temperatures. The points represent- ing K ∗ may appear to lie on the z -axis but they are are slightly below it. In the 1D-case we suspected that there is a limit curve for the high-temperature region, but we suspect that the exact data that produced this part of the plot rely on far too small lattices to give any con- clusive evidence. Also, the p, q -find algorithm is rather sensitive to noise in this region to be useful for sampled data. However, as we said before, this problem goes away once K ≥ K ∗ . Figure 12 shows y and z versus K for all the lattices though for the sampled data we only show low-temperature data. The red line is located at K$_{c}$ = atanh ( √$_{2}$ − 1 ) ≈ 0 . 44068. y z FIG. 11: Left: z = n ( q − 1) vs log n at K ∗ for the L × L -lattice, L = 6, 8, 10, 12, 14, 16, 32, 64, 128, 256, 512. The curve is 3 − 1 . 5 log n − 6 log log n . Right: y = n ( p − 1) vs z = n ( q − 1) for the L × L -lattice, L = 6, 8, 10, 12, 14, 16, 32, 64, 128, 256, 512 (512 barely visible near the z -axis). The points represent K ∗ .