content_image/355976.jpg content_image/355975.jpg
and for ℓ = 1 this simplifies to 

Compare this with (66). We thus have a = − 2 e − 2 $^{K}$. Now y and z are related as y = 2 w − a where w is defined by (68). If we set K = 2 / 15 then a = − 1 . 531857, and choosing z = − 3 . 537 indeed gives us y = 1 . 01002. To actually solve z as a function of K seems harder though. However, numerical experimentation suggests that y and z for small K behave as 

where c$_{1}$ ≈ 6 . 164 and c$_{2}$ ≈ − 10 . 33. 

<img src='content_image/355976.jpg'>

<img src='content_image/355975.jpg'>

Strangely, when the p, q -distribution fit the magnetisation distribution so well one might think that the free energy would be well approximated by (139). This is not so. The p, q -approximation differs clearly from the asymptotic free energy, given by log (2 cosh K ). 

## B. 2D-lattices 

For the 2-dimensional lattices we can rely on exact data only for up to L = 16 and they were computed according to the method in $^{14}$. We have sampled data for L = 32 , 64 , 128 , 256 , 512, collected with the methods described in 15 and $^{16}$. These meth- ods gave us the energy distribution and then it is just a matter of combining this with the distribution of magnetisations for each given energy as described in $^{12}$. Figure 10 shows an example of some distributions for the 128 × 128-lattice together with their fitted p, q -binomial distributions. The fit is fairly good, but hardly excellent near K ∗ . However, as the figure shows, at K = 0 . 4388 (i.e. for L = 128) the fit is practically spot on. For the lattices we have studied there is always one such temperature where the p, q -distribution fit particularly well. This point is located between K ∗ and K$_{c}$ and is very close to, but not exactly equal to, the point where the susceptibility is at its maximum. 

Of course, for high temperatures (small K ) and low temperatures (high K ) the fit is typ- ically very good but in the high-temperature region the measured y and z are unfortunately extremely sensitive to noise. As we get closer to the critical region where the distribution becomes bimodal this problem goes away, even though the sampled distributions are more 

