and Had we let λ$_{2}$ = 0, instead of λ$_{2}$ = − 6, then we would have ended up with a factor log 3 / 4 n in the denominator of (108) and a factor log 3 / 2 n in the denominator of (109). For the second case we choose λ$_{1}$ = 0, λ$_{2}$ < 0 and λ$_{3}$ = − 6. We get This together with (101) gives us and Γ( 1 / 4 ) ( − λ$_{2}$ ) e These expressions obviously converge extremely slowly and are probably not of any use for n that might occur in practical situations. We have managed to verify (108) and (109) by using the method described in section VI, again using Mathematica, only in the special case z = − log n , i.e. λ$_{1}$ = − 1, λ$_{0}$ = λ$_{2}$ = λ$_{3}$ = 0. We could then confirm that Computing the moments of this distribution produces the same result as setting λ$_{1}$ = − 1 and λ$_{0}$ = λ$_{2}$ = λ$_{3}$ = 0 in (106) and then computing the moments in the same the way we obtained (108) and (109). A more general computation seems not to be within reach with our current set of tools though. To conclude this section we note that the exponent of n in (113) is 3 / 4 − λ$_{1}$/ 8. For this exponent to stay less than one we thus need λ$_{1}$ > − 2, giving us a bound on z . A state τ on a graph G is a function from the set of vertices to {± 1 } . There are thus 2 n states for a graph on n vertices. We define the energy of a state as E ( τ ) = ∑ ij τ$_{i}$τ$_{j}$ where the sum is taken over all edges ij of G . The magnetisation is defined as M ( τ ) = ∑ i τ$_{i}$ with 28 X. THE ISING MODEL R$_{p,q}$ ( n, n/ 2 , x n 7 / 8 log 3 / 4 n ) ∼ exp ( − x 4 12 ) (113) σ$_{1}$ ∼ 3 1 / 4 √$_{2}$$_{π}$ Γ( 1 / 4 ) n 3 / 4 log − λ$_{2}$/ 8 n ( − λ$_{2}$ ) 3 / 4 e λ$_{0}$/ 8 (111) σ$_{2}$ ∼ $_{2}$√$_{6}$$_{π}$ 1 / 4 2 n 3 / 2 log − λ$_{2}$/ 4 n ( λ$_{2}$ ) 3 / 2 e λ$_{0}$/ 4 (112) Φ( z ) ∼ ( − λ$_{2}$ ) 3 12 e λ$_{0}$/ 2 log λ$_{2}$/ 2 n (110) σ$_{2}$ ∼ $_{2}$√$_{6}$$_{π}$ Γ( 1 / 4 ) 2 n 3 / $^{2}$− λ$_{1}$/ 4 ( − λ$_{1}$ ) 3 / 2 e λ$_{0}$/ 4 (109)