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 F0ex0ex=0ex0ex1. 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 0. 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.