for a graph on n vertices and m edges and it is here implied that p and q depend on K . Why this expression? Note that Z 0 = a ( m,n ) e K m = e K m and thus we have from which the result follows. Compare with (121) where this relation holds exactly. Ac- tually we expect (139) to be a good approximation for K near 0, where the distributions are close to binomial, and for very high K where all the probability mass is concentrated on the extreme magnetisations. In the left plot of figure 5 we show the exactly computed free energy (red curve) for a complete bipartite graph on 16 + 16 vertices together with the p,q -approximation (139) (points). The fit is indeed very good for the whole temperature range. Taking a derivative of the points with respect to K produces a good approximation to the internal energy as the right plot shows. The plots in figure 4 are very representative for several graphs of interest. We intend to focus on the graphs that traditionally are studied in statistical physics; lattice graphs. We will take a look at the simple lattices in 1, 2, 3, 4 and 5 dimensions. Just to be clear, a 1D-lattice is a cycle C$_{n}$ on n vertices and it is 2-regular, i.e. 2 neighbours for each vertex. The 2-dimensional L × L -lattice is the cartesian product of two cycles on L vertices. The 34 XI. LATTICES FIG. 5: Left: free energy (red curve) compared to the formula (139) (points) for K$_{16}$$_{,}$$_{16}$ . Right: internal energy (red curve) compared to the derivative of the points produced by (139) (points) for K$_{16}$$_{,}$$_{16}$ . P$_{p,q}$ ( n, 0) = [ n 0 ] p,q Ψ$_{p,q}$ ( n ) = 1 Ψ$_{p,q}$ ( n ) = Z 0 = e K m Z (140) 0.00 0.02 0.04 0.06 0.08 0.10 K 0.70 0.75 0.80 0.85 F 0.00 0.02 0.04 0.06 0.08 0.10 K 0 1 2 3 4 5 6 $^{GLYPH<PartialDiff>}$F GLYPH<Slash1> $^{GLYPH<PartialDiff>}$K