product thus has n = L 2 vertices and it is 4-regular. The d -dimensional L × L ×· · ·× L -lattice is a product of d cycles on L vertices, thus having a total of n = L d vertices. It is obviously 2 d -regular. Assuming finite-size scaling to hold then for a d -dimensional lattice we have and correspondingly for the second moment Note also that for an r -regular triangle-free graphs we have P ( M = n ) = P ( M = n − 2) when K = log n 2 r . For 1D-lattices we can compute the coefficients a ( E, M ) exactly. It is an exercise to show that the number of states with k negative spins and ℓ negative spin products (over the edges) is where M = n − 2 k and E = n − 2 ℓ . The distribution of magnetisations do not behave in a way representative for lattices of higher dimension. However, for extremely low temperatures the probabilities P ( M = − n ) = P ( M = n ) will dominate the other probabilities. The two outermost probabilities, P ( M = n ) and P ( M = n − 2), are equal when K = log n 4 . For the 1D-lattice the distribution is here sharply unimodal, while for higher dimensions the distribution is bimodal and has its peaks at the extreme magnetisations. For K larger than (log n ) / 4 the distribution actually has three peaks, i.e. a local maximum at M = 0. For 1D-lattices the p, q -approximation of the distribution thus breaks down beyond this K since it can not model a local maximum in the middle as well as peaks at the ends; they are at most bi modal. For K less than this point the p, q -distribution is a very good approximation. Figure 6 demonstrates this clearly; for the flattest distribution (low temperature) the fitted p, q -distribution starts to deviate from the actual distribution. In figure 7 we plot y = n ( p − 1) and z = n ( q − 1) versus K for a range of different n . Clearly there is some limit curve here, though we have not established what the limit function is. In figure 8 we see y = n ( p − 1) versus z = n ( q − 1) for different n . The right plot of figure 8 shows the value 35 a ( E, M ) = ( k k − ℓ/ 2 )( n − k − 1 n k ℓ/ 2 ) + ( k − 1 k ℓ/ 2 )( n − k n − k ℓ/ 2 ) (143) A. 1D-lattices σ$_{2}$ = n χ/ 4 ∝ n L γ/ν = n ( n 1 $^{/d}$) γ/ν = n 1+ γ/d ν (142) σ$_{1}$ = n ¯ µ/ 2 ∝ n L − β/ν = n ( n 1 $^{/d}$) − β/ν = n $^{1}$− β/d ν (141)