In this paper 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 graphs, i.e., graphs where a random walker returns to its starting point with an average probability $\overline{F}<1.\mathrm{}$ This result, which is here proven for models with $O\left(n\right)\mathrm{}$ 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.

• Received 16 February 1999

DOI:https://doi.org/10.1103/PhysRevE.60.1500