Transience on the average and spontaneous symmetry breaking on graphs

We give a rigorous proof of 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 (F) over bar < 1. The proof holds for models with O(n) symmetry with n greater than or equal to 1, therefore including the Ising model as a particular case. This result, together with the generalized Mennin-Wagner theorem, completes the picture of phase transitions for continuous symmetry models on graphs and leads to a natural classification of general networks in terms of the two geometrical superuniversality classes of recursive on the average and transient on rite average.