content_image/358088.jpg content_image/358087.jpg content_image/358085.jpg content_image/358086.jpg content_image/358084.jpg
and 

<img src='content_image/358088.jpg'>

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 ) 2 ( − λ$_{2}$ ) 3 / 2 e λ$_{0}$/ 4 These expressions obviously converge extremely slowly and are probably not of any use for n that might occur in practical situations. 

<img src='content_image/358087.jpg'>

<img src='content_image/358085.jpg'>

<img src='content_image/358086.jpg'>

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 

<img src='content_image/358084.jpg'>

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 . 

## X. THE ISING MODEL 

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 

