PHASE-TRANSITIONS AND RANDOM-WALKS ON GRAPHS - A GENERALIZATION OF THE MERMIN-WAGNER THEOREM TO DISORDERED LATTICES, FRACTALS, AND OTHER DISCRETE STRUCTURES

It is rigorously proven that classical O(n) and quantum spin-s Heisenberg ferromagnetic models on generic networks cannot have spontaneous magnetization at any finite temperature if random walks on the same structure are recursive. It follows as a corollary that on fractals a continuous symmetry can be spontaneously broken only if the spectral dimension is greater than 2.