The second case implies that every positive spin lies inside some border, since it must lie on the opposite side of the negative spin which is always external. We now restrict our attention to the configurations belonging to the first class, denoting by a subscript N the quantities that pertain to it; we can obtain a good estimate of the number of negative spins, 〈 N$_{−}$ 〉$_{N}$ , as follows: the sign of a spin p is negative if it is contained inside an odd number of borders, positive otherwise; we obtain a very naive, yet effective, approximation if we consider any spin contained inside at least one border as negative: letting I p be 1 if p is inside at least a border, 0 otherwise, we can write We are now to give a reasonable estimate of 〈 I $^{p}$〉 : take all the configurations C with at least one border containing p , call b$_{min}$ the length of the shortest border in C containing p and let k be the number of borders containing p , so as to write Now fix b$_{min}$ and consider the configurations containing k borders: if we remove the shortest border from such a configuration C , we obtain a new configuration C ′ with ( k − 1) borders containing p , each of them at least b$_{min}$ long. C ′ will be present in the partition function Z , but different configurations with k borders C may give the same C $^{′}$; defining now µ $^{p}$( b ) as the number of possible borders of b links containing p , the degeneration induced by removing the shortest border is not greater than µ $^{p}$( b$_{min}$ ). The energy of a configuration C and the corresponding Boltzmann factor obey if we now limit the sum in the partition functions to those configurations obtained by removing a border from the numerator, we can write In this way the average number of minus spins is bounded by a function depending only on the number of borders encircling a given spin: E$_{C}$ ≥ E$_{C}$ ′ + 2 βJ$_{min}$b$_{min}$, e − β H C ≤ e −H$_{C}$ ′ e − 2 βJ$_{min}$$^{b$_{min}$}$; Z − 1 ∑ k ≥ 1 ∑ C|$_{k borders}$ e − β H ≤ µ $^{p}$( b$_{min}$ ) e − 2 βJ$_{min}$$^{b$_{min}$}$. 〈 N$_{−}$ 〉$_{N}$ ≤ ∑ p ∈G ∑ b$_{min}$ µ $^{p}$( b$_{min}$ ) e − 2 βJ$_{min}$$^{b$_{min}$}$; 8 〈 I $^{p}$〉 = Z − 1 ∑ C|$_{p inside}$ e − β H = Z − 1 ∑ b$_{min}$ ≥ 1 ∑ k ≥ 1 ∑ C| b min k borders e − β $^{H}$. 〈 N$_{−}$ 〉$_{N}$ ≤ ∑ p ∈G 〈 I $^{p}$〉 .