The spectral dimension and geometrical universality on graphs

The spectral dimension d̄ of an infinite graph, defined according to the asymptotic behavior of the Laplacian operator spectral density, seems to be the right generalization of the Euclidean dimension d of lattices to non translationally invariant networks when dealing with dynamical and thermodynamical properties. In fact d̄ exactly replaces d in most laws where dimensional dependence explicitly appears: the spectrum of harmonic oscillations, the average autocorrelation function of random walks, the critical exponents of the spherical model, the low temperature specific heat, the generalized Mermin-Wagner theorem, the infrared singularities of the Gaussian model and many other. Still, d̄ would be a rather unsatisfactory generalization of d if it hadn't a second fundamental property: the independence of geometrical details at any finite scale (or geometrical universality). Here we show that d̄ is invariant under all geometrical transformations affecting only finite scale topology. In particular we prove that d̄ is left unchanged by any quasi-isometry (including coarse-graining and addition of finite range couplings), by local rescaling of couplings and by addition of infinite range of couplings provided they decay faster than a given power law.