Diffusion, localization and dispersion relations on ``small-world'' lattices

TL;DR

This paper investigates the spectral properties of the Laplacian on small-world lattices, combining analytical and numerical methods to understand localization, dispersion, and density of states.

Contribution

It introduces a transfer matrix formalism with a self-consistent potential and applies an effective medium approach to analyze spectral features on small-world structures.

Findings

Effective medium calculation matches numerical density of states

Localization effects are quantitatively described by single defect approximation

Dispersion relations are derived for eigenmodes in the extended spectrum

Abstract

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 a 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.