content_image/372998.jpg
<img src='content_image/372998.jpg'>

is proved that h ( n ) ≥ log n is both a sufficient and a necessary condition for spontaneous magnetization. 

## DISCUSSION 

To give a more intuitive interpretation of the theorem we proved, we can proceed as follows: if the number of borders grows less than exponentially, we can argue that all of these borders will contain a number of spins increasing slowly with the length of the border; as a consequence, the formation of large clusters of spins in a magnetized graph will be energetically unfavoured, so that the latter will result a stable state. On the other hand, if µ $^{i}$( b ) grows very fast with b , we expect that some of the borders will be far from the vertex i , so that more and more vertices will be enclosed in short (low b ) borders; this in turn means that large clusters of spins can be flipped spending a small amount of energy, so that a magnetized graph may be unstable with regard to thermal fluctuations. 

The condition of our theorem is a strong one, in that it investigates a global property of the graph. For this reason it can not be a necessary condition for achieving spontaneous magnetization: if a graph has a part, which has zero measure in the thermodynamic limit, 

