0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120 K $^{GLYPH<Minus>}$9.4 $^{GLYPH<Minus>}$9.2 $^{GLYPH<Minus>}$9.0 $^{GLYPH<Minus>}$8.8 $^{GLYPH<Minus>}$8.6 $^{GLYPH<Minus>}$8.4 z 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120 K $^{GLYPH<Minus>}$0.20 $^{GLYPH<Minus>}$0.18 $^{GLYPH<Minus>}$0.16 $^{GLYPH<Minus>}$0.14 $^{GLYPH<Minus>}$0.12 $^{GLYPH<Minus>}$0.10 $^{GLYPH<Minus>}$0.08 y FIG. 21: Le ft: z = n ( q − 1) vs K with K > K ∗ for the L × L × L × L × L -lattice, L = 4 , 6 , 8 , 10 , 12 (downwards ). Right: y = n ( p − 1) vs K with K > K ∗ for the L × L × L × L × L -lattice, L = 4 , 6 , 8 , 1 0 , 12 (upwards). In both plots the red line indicates location of K$_{c}$ and the points are the location s of K ∗ . Analogous ly, the second cumulant (normalised) should behave as where the ϱ$_{m}$ were defined in section VIII. Note that for a fixed z the ϱ$_{m}$ now depend only on a . The third and fourth cumulants of the absolute magnetisation, divided by respectively 8 n 9 / 4 and 16 n $^{3}$, quite analogously approach their corresponding limits and Through a simple scaling analysis based on our sampled data we have found that the nor- malised third cumulant has a limit maximum of about 0 . 0205 and a minimum of − 0 . 0500. The fourth normalised cumulant has a limit maximum of 0 . 0229 and a minimum of − 0 . 0278, based upon our sampled data. Choosing z = − 9 . 87 puts the maximums and minimums of the limit curves at appropriate values. Now we identify the coupling K where the minimum of the fourth cumulant occurs with the point a where the minimum of the corresponding limit curve occurs and likewise for the maximum, thus providing us with a rescaling trans- lating K into a . In figure 22 and 23 the first four cumulants are shown together with their estimated limit curves for z = − 9 . 87. Indeed the red curve may provide us with a limit. Given a lattice size L we denote by K$_{min}$ ( L ) the location of the minimum fourth cumulant and by K$_{max}$ ( L ) the location of the maximum. Analogously for the limit curve, given a z we 47 ϱ$_{4}$ ϱ$_{0}$ − 4 ϱ$_{1}$ σ$_{3}$ ϱ 2 0 − 3 σ 2 ϱ 2 0 + 12 ϱ 2 1 ϱ$_{2}$ ϱ 3 0 − 6 ϱ 4 1 ϱ 4 0 (152) ϱ$_{3}$ ϱ$_{0}$ − 3 ϱ$_{1}$ σ$_{2}$ ϱ 2 0 + 2 ϱ 3 1 ϱ 3 0 (151) σ$_{2}$ − σ 2 1 n 3 / 2 → ϱ$_{2}$ ϱ$_{0}$ (150)