GEOMETRICAL UNIVERSALITY IN VIBRATIONAL DYNAMICS

A good generalization of the Euclidean dimension to disordered systems and noncrystalline structures is commonly required to be related to l arge scale geometry and it is expected to be independent of local geom etrical modifications. The spectral dimension, defined according to th e low frequency density of vibrational states, appears to be the best candidate as far as dynamical and thermodynamical properties are conce rned. In this letter we give the rigorous analytical proof of its inde pendence of finite scale geometry. We show that the spectral dimension is invariant under local rescaling of couplings and under addition of finite range couplings, or infinite range couplings decaying faster t hen a characteristic power law. We also prove that it is left unchange d by coarse graining transformations, which are the generalization to graphs and networks of the usual decimation on regular structures. A q uite important consequence of all these properties is the possibility of dealing with simplified geometrical models with nearest-neighbors i nteractions to study the critical behavior of systems with geometrical disorder.