THE SPECTRAL DIMENSION AND GEOMETRICAL UNIVERSALITY ON GRAPHS

The spectral dimension (d) over bar of an infinite graph, defined acco rding to the asymptotic behavior of the Laplacian operator spectral de nsity, 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) over bar ex actly replaces d in most laws where dimensional dependence explicitly appears: the spectrum of harmonic oscillations, the average autocorrel ation function of random walks, the critical exponents of the spherica l model, the low temperature specific heat, the generalized Mermin-Wag ner theorem, the infrared singularities of the Gaussian model and many other. Still, (d) over bar would be a rather unsatisfactory generaliz ation of d if it hadn't a second Fundamental property: the independenc e of geometrical details at any finite scale (or geometrical universal ity). Here we show that (d) over bar is invariant under all geometrica l transformations affecting only finite scale topology. In particular we prove that (d) over bar is left unchanged by any quasi-isometry (in cluding coarse-graining and addition of finite range couplings), by lo cal rescaling of couplings and by addition of infinite range of coupli ngs provided they decay faster than a given power law.