J. Phys. IV France
Volume 08, Numéro PR6, October 1998
International Conference on Disorder and Chaos in honour of Giovanni Paladin
Page(s) Pr6-81 - Pr6-85
International Conference on Disorder and Chaos in honour of Giovanni Paladin

J. Phys. IV France 08 (1998) Pr6-81-Pr6-85

DOI: 10.1051/jp4:1998610

The spectral dimension and geometrical universality on graphs

R. Burioni and D. Cassi

Dipartimento di Fisica, INFM and INFN, Università di Parma, viale delle Scienze, 43100 Parma, Italy

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.

© EDP Sciences 1998