Random walks and localization on the Penrose lattice

We study the low energy limit of the integrated density of states for the standard tight-binding model on the Penrose lattice. A conjecture is presented for the existence of the so-called harmonic coordinates; it implies a central limit theorem for random walks. A simple approximation scheme is proposed to compute the diffusion constant. In a second part of the paper, we consider the effect of a diagonal term in the tight-binding hamiltonian, suppressing all local symmetries. This self-similar potential provides a simple mechanism for localization; the existence of gaps in the spectrum is demonstrated. (C) 2000 Elsevier Science B.V. All rights reserved.