The spectral dimension and geometrical universality on graphs

Abstract
The spectral dimension d^- 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 d^- 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, d^- 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 d^- is invariant under al1 geometrical transformations affecting only finite scale topology. In particular we prove that d^- 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.