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 p(ω) ~ω^{d_s -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.