https://doi.org/10.1007/s100510051038
Diffusion, localization and dispersion relations on "small-world" lattices
CNRS - Laboratoire de Physique Théorique de
l'ENS (LPTENS is a unité mixte du CNRS,
associée à l'École Normale
Supérieure et à l'Université de Paris-Sud.) ,
24 rue Lhomond, 75231 Paris Cedex 05, France
Received:
23
March
1999
Published online: 13 August 2012
The spectral properties of the Laplacian operator on "small-world" lattices, that is mixtures of unidimensional chains and random graphs structures are investigated numerically and analytically. A transfer matrix formalism including a self-consistent potential à la Edwards is introduced. In the extended region of the spectrum, an effective medium calculation provides the density of states and pseudo relations of dispersion for the eigenmodes in close agreement with the simulations. Localization effects, which are due to connectivity fluctuations of the sites are shown to be quantitatively described by the single defect approximation recently introduced for random graphs.
PACS: 63.50.+x – Vibrational states in disordered systems / 71.23.An – Theories and models; localized states / 71.55.Jv – Disordered structures; amorphous and glassy solids
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 1999
