Local vs average behavior on inhomogeneous structures: Recurrence on the average and a further extension of Mermin-Wagner theorem on graphs

Spontaneous breaking of a continuous symmetry cannot occur on a recursive structure, where a random walker returns to its starting point with probability F = 1. However, some examples showed that the inverse is not true. We explain this by further extension of the previous theorem: Indeed, even if F < 1 everywhere, its average over all the points can be 1. We prove that even on these recursive on the average structures the average spontaneous magnetization of O(n) and Heisenberg models is always O. This difference between local and average behavior is fundamental in inhomogeneous structures and requires a ''doubling'' of physical parameters such as spectral dimension and critical exponents.