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this result states that, no matter what the maximum number of spins you can isolate inside a border is, as long as µ $^{p}$( b ) grows at most exponentially the value of 〈 N$_{−}$ 〉$_{N}$ can be limited at low enough temperatures. An analogous result holds for configurations of class P when exchanging the roles of positive and negative spins: 

Let now µ ( b )) = sup$_{p}$ µ $^{p}$( b ). We can now prove the following theorem: 

<img src='content_image/331157.jpg'>

Theorem. If on an infinite graph µ ( b ) ≤ A $^{b}$, definitely for b ≥ ¯ b and for some for some A > 0 , then the graph exhibits spontaneous magnetization at large enough β (low enough temperatures). 

Proof. The average modulus of magnetization is | M | = N − $^{1}$( N$_{+}$ − N$_{−}$ ); writing the Boltz- mann factor for a configuration C as P$_{C}$ = Z − $^{1}$e − β H ( C $^{)}$, we can write the following: 

<img src='content_image/331161.jpg'>

When the sum on the last line converges, the equation tells us that, for large enough β (low temperatures), 〈| M |〉 is greater than a positive constant, so that spontaneous magnetization on an infinite graph is achieved, while in general 〈| M |〉 is finite for every N , but can tend to zero as N → ∞ . 

The main problem in employing the previous theorem is determining bounds on µ ( b ). The simplest case in which the hypothesis does not hold is a situation in which for some finite b the number of borders surrounding a given point is infinite. As a sound check of the 

