SCALING AND SCALING-RELEVANT DYNAMICAL PROPERTIES OF PENROSE TILING

The scaling and the scaling-relevant spectral properties of Penrose tiling are investigated. The fractal dimension d(f) of this tiling is analytically obtained, which is two, equal to its Euclidean dimension. Similar to usual self-similar structure, the vibrational density of states for Penrose lattice is also found to follow a power law rho(omega) similar to omega(ds-1) with spectral dimension d(s) = 2, which accounts for a special vibrational excitation in quasicrystals: the fracton-like excitation, whose state is critical. The simulation of random walk on this Penrose lattice indicates that the diffusive dimension d(w) = 2, thus the relation d(s) = 2d(f)/d(w) holds.