moment exponent would be 53 / 64 = 0 . 828125 and 53 / 32 = 1 . 65625 for the second moment, slightly below the lower bound of the estimate intervals. Choosing λ$_{1}$ = − 2 / 3 would give exponents 5 / 6 = 0 . 8333 . . . and 5 / 3 = 1 . 666 . . . respectively, slightly above the upper bound of the intervals. Let us suggest, as an example, that λ$_{0}$ = 6 . 8, λ$_{1}$ = − 2 / 3, λ$_{2}$ = − 6 and λ$_{3}$ = 0 in the expression (109). In figure 14 the curve use these parameters for z at K ∗ , i.e. z = 6 . 8 − (2 / 3) log n − 6 log log n . Will the points eventually converge to the curve? It would take considerably larger lattices to shed any light on this. We also have the problem what λ$_{0}$ should be. Using λ$_{0}$ = 6 . 8 means that the coefficient in (109) is about 0 . 393. Comparing the measured σ$_{2}$ with n 5 / 3 gives a factor of roughly 0 . 16 though the data are certainly far from conclusive. Since the distribution fit is not perfect a different constant is perhaps to be expected. Also, slow convergence is to be expected here. The right plot of figure 14 shows y versus z for K > K ∗ . Note the peculiar backwards movement of z getting more and more pronounced for larger L . The curves for 16, 32 and 64 show signs of approaching some limit curve. We don’t have data for very low temperatures for the smaller lattices though, except for L = 4. The plots in figure 15 shows y and z versus K for K > K ∗ . The red lines show location of K$_{c}$ ≈ 0 . 2216546, found in $^{12}$, but see also20 for a theoretical estimate of K$_{c}$ . 42 FIG. 14: Left: z = n ( q − 1) vs log n at K ∗ for the L × L × L -lattice, L = 4 , 6 , 8 , 12 , 16 , 32 , 64. The line though the points is − 5 . 3 − log n and the curve is 6 . 8 − (2 / 3) log n − 6 log log n . Right: y = n ( p − 1) vs z for the L × L × L -lattice, L = 4 , 6 , 8 , 12 , 16 , 32 , 64 (leftwards) for K > K ∗ . Higher values of K when we move downwards left. The red curve is y = 2 w with w defined by (59). 4 6 8 10 12 14 log n $^{GLYPH<Minus>}$18 $^{GLYPH<Minus>}$16 $^{GLYPH<Minus>}$14 $^{GLYPH<Minus>}$12 $^{GLYPH<Minus>}$10 $^{GLYPH<Minus>}$8 $^{GLYPH<Minus>}$6 $^{GLYPH<Minus>}$4 z $^{GLYPH<Minus>}$18 $^{GLYPH<Minus>}$16 $^{GLYPH<Minus>}$14 $^{GLYPH<Minus>}$12 $^{GLYPH<Minus>}$10 z $^{GLYPH<Minus>}$0.30 $^{GLYPH<Minus>}$0.25 $^{GLYPH<Minus>}$0.20 $^{GLYPH<Minus>}$0.15 $^{GLYPH<Minus>}$0.10 $^{GLYPH<Minus>}$0.05 y