• internal if – B$_{i}$ is closed and A$_{i}$ ∩ E =  , or – B$_{i}$ is open and A$_{i}$ contains fewer elements than C$_{i}$ , or – B$_{i}$ is open, A$_{i}$ has the same size as C$_{i}$ and the points in A$_{i}$ linked to B$_{i}$ have negative spin; • external if – B$_{i}$ is closed and A$_{i}$ ∩ E ̸ =  , or – B$_{i}$ is open and A$_{i}$ has more elements than C$_{i}$ , or – B$_{i}$ is open, A$_{i}$ has the same size as C$_{i}$ and the points in A$_{i}$ linked to B$_{i}$ have positive spin. The reason why we had to select a finite subgraph G$_{N}$ is that we need to be able to count the number of spins in the graph for the previous definitions to make sense. Let now σ$_{i}$ = ± 1 be a spin variable for each vertex i ∈ P . We define the Ising Hamil- tonian on a graph as THE FERROMAGNETIC ISING MODEL ON A GRAPH H = − ∑ ( i,j ) ∈P×P J$_{ij}$σ$_{i}$σ$_{j}$ − ∑ i ∈P σ$_{i}$h$_{i}$, (1) where the couplings J$_{ij}$ = J$_{ji}$ must satisfy 0 ≤ J$_{ij}$ < J$_{Max}$ < ∞ for some J$_{Max}$ , and J$_{ij}$ > 0 if and only if ( i, j ) ∈ L . In the following we will set the external field to zero everywhere ( h$_{i}$ ≡ 0). Now that we have presented the terminology we’ll be using, we are going to study the equilibrium statistical mechanics of the Ising model at inverse temperature β , and in par- ticular the modulus of the magnetization 〈| M |〉 = Z − 1 ∑ { σ$_{i}$ } ∣ ∑ j ∈P σ$_{j}$ ∣ |P| e − β H ( { σ$_{i}$ } $^{)}$, (2) where Z = ∑ { σ$_{i}$ $_{}}$e − β H ( { σ$_{i}$ } ) is the partition function, and |P| is the cardinality of P . 5