Random walks and localization on the Penrose lattice

We study the low energy limit of the integrated density of states for the s tandard tight-binding model on the Penrose lattice. A conjecture is present ed for the existence of the so-called harmonic coordinates; it implies a ce ntral limit theorem for random walks. A simple approximation scheme is prop osed to compute the diffusion constant. In a second part of the paper, we c onsider the effect of a diagonal term in the tight-binding hamiltonian, sup pressing all local symmetries. This self-similar potential provides a simpl e mechanism for localization; the existence of gaps in the spectrum is demo nstrated. (C) 2000 Elsevier Science B.V. All rights reserved.