LOCAL VS AVERAGE BEHAVIOR ON INHOMOGENEOUS STRUCTURES - RECURRENCE ONTHE AVERAGE AND A FURTHER EXTENSION OF MERMIN-WAGNER THEOREM ON GRAPHS

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