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: 〈 N$_{+}$ 〉$_{P}$ ≤ ∑ p ∈G ∑ b$_{min}$ µ $^{p}$( b$_{min}$ ) e − 2 βJ$_{min}$$^{b$_{min}$}$. Let now µ ( b )) = sup$_{p}$ µ $^{p}$( b ). We can now prove the following theorem: 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: 〈| M |〉 = ∑ C∈P∪N | M$_{C}$ | P$_{C}$ = ∑ C∈ ( P∪N ) + M$_{C}$ P$_{C}$ − ∑ C∈ ( P∪N ) − M$_{C}$ P$_{C}$ = ( ∑ C∈N + M$_{C}$P$_{C}$ − ∑ C∈N − M$_{C}$P$_{C}$ ) + ( ∑ C∈P + M$_{C}$P$_{C}$ − ∑ C∈P − M$_{C}$P$_{C}$ ) = ( ∑ C∈N M$_{C}$P$_{C}$ − ∑ C∈P M$_{C}$P$_{C}$ ) + 2 ( ∑ C∈P + M$_{C}$P$_{C}$ − ∑ C∈N − M$_{C}$P$_{C}$ ) ≥ ( ∑ C∈N M$_{C}$P$_{C}$ − ∑ C∈P M$_{C}$P$_{C}$ ) = 1 − 2 N ( ∑ C∈N ( N$_{−}$ )$_{C}$ P$_{C}$ + ∑ C∈P ( N$_{+}$ )$_{C}$ P$_{C}$ ) ≥ 1 − 4 N ∑ p ∈ V ∑ b$_{min}$ µ $^{p}$( b$_{min}$ ) e − 2 βJ$_{min}$b$_{min}$ ≥ 1 − 4 e − 2 βJ$_{min}$ ∑ b ≥ 1 µ ( b$_{min}$ ) e − 2 β ( b − 1) $^{J$_{min}$}$, 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 9