and R ± j are disconnected; moreover we require that R ± ≡ R ± 1 ∪ R ± 2 ∪ · · · ∪ R ± $_{S}$, R ± i ∩ R ± j =  ∀$_{i}$$_{̸}$$_{=}$$_{j}$ . We are selecting individual clusters of homogeneous spins, so satisfying the above requisites is always possible. Setting now ∂R ± i ≡ { ( a, b ) ∈ L : a ∈ R ± i , b / ∈ R ± i } , ± the sets B ± ≡ ∂R ± 1 ∪ · · · ∪ ∂R ± S are a collection of links each defined unambiguously, and furthermore B + ≡ B $^{−}$. It may happen that for some i the subgraph G \ R ± i is made of two disconnected subgraphs (e.g. when a ring of plus spins is surrounded by minus spins); as a consequence ∂R ± i is not a border according to our definition. In that case it is possible to split ∂R ± i into subsets, so that each of them divides G into two connected subgraphs. After dealing in this way whith all the R ± i , we are left with a collection of well-defined borders ∂T + j , with j = 1 , . . . , U$_{+}$ , and ∂T − k , with k = 1 , . . . , U$_{−}$ where U$_{+}$, U$_{−}$ > S . It is still possible that some of the borders T + i , while defining exactly the same zones, have no correspective in T − i but, since they nevertheless verify B + ≡ B $^{−}$, they belong to the same border class, completing the proof. The main consequence of this result is that we can substitute a sum over border classes for a sum over configurations whenever needed, and we can infer from the structure of the borders some limiting properties for the spins distributions, as we’ll see soon. It is worthwhile to explicitly notice that, when we pass from a sum over configurations to one over borders, and not border classes, we overcount some borders, as there are more than one representative of each border class: this is not going to be a problem in the use we’ll make of this result. We can divide the set of all configurations on G$_{N}$ into two classes: GENERALIZED PEIERLS-GRIFFITHS’ THEOREM • all the negative spins are internal to some border (class N ), • at least a negative spin exists that is external to all borders (class P ). 7