noisy there. Regarding the free energy it is well-fitted by (139) for low temperatures K > K ∗ though less well for high temperatures K < K ∗ . $^{GLYPH<Minus>}$6000 $^{GLYPH<Minus>}$4000 $^{GLYPH<Minus>}$2000 2000 4000 6000 0.00005 0.00010 0.00015 0.00020 0.00025 FIG. 10: Magnetisation distributions for the 128 × 128-lattice (red) together with fitted p, q -binomial distributions P$_{p,q}$ ( n, k ) (blue) vs k − n/ 2 at K = 0 . 432, K ∗ = 0 . 43467, K = 0 . 437, K = 0 . 4388 and K$_{c}$ = 0 . 44068 (downwards at y -axis). Recall from section IX how the exponents of the moment growth rates could be computed if we allow z to depend on n . For the 2D-lattices it is known that β = 1 / 8, γ = 7 / 4 and ν = 1, see $^{17}$,18 and $^{19}$. Thus the first moment σ$_{1}$ should scale as n 15 / 16 and the second moment σ$_{2}$ as n 15 / $^{8}$. From equation (108) and (109) this would be achieved by choosing λ$_{1}$ = − 3 / 2, λ$_{2}$ = − 6 and λ$_{3}$ = 0. The left plot of Figure 11 shows z versus log n at K ∗ together with the curve 3 − 1 . 5 log n − 6 log log n . The constant λ$_{0}$ is chosen only to make the curve look plausibly near the points. The point for L = 512 deviate slightly but we suspect that noise in the sampled data explains this. With λ$_{0}$ = 3 the coefficient of n 15 / 8 obtained from (109) would be 0 . 301 though the measured σ$_{2}$ divided by n 15 / 8 are closer 0 . 08. To get this we have to choose λ$_{0}$ ≈ 8 . 3. In that case the convergence is extremely slow. Note also that the fitted p, q -distribution is far from perfect which would contribute some amount of error as well. 39