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.