The inverse Mermin-Wagner theorem for classical spin models on graphs

In this letter we present the inversion of the Mermin-Wagner theorem on graphs, by proving the existence of spontaneous magnetization at finite temperature for classical spin models on transient on the average (TOA) graphs, i.e. graphs where a random walker returns to its starting point with an average probability $\bar F < 1$. This result, which is here proven for models with O(n) symmetry, includes as a particular case $n=1$, providing a very general condition for spontaneous symmetry breaking on inhomogeneous structures even for the Ising model.