Scaling analysis of stationary probability distributions of random walks on one-dimensional lattices with aperiodic disorder

Stationary probability distributions of one-dimensional random walks on lattices with aperiodic disorder are investigated. The pattern of the distribution is closely related to the diffusional behavior, which depends on the wandering exponent $\Omega$ of the background aperiodic sequence: If $\Omega<0$, the diffusion is normal and the distribution is extended. If $\Omega>0$, the diffusion is ultraslow and the distribution is localized. If $\Omega=0$, the diffusion is anomalous and the distribution is singular, which shows its complex and hierarchical structure. Multifractal analysis are performed in order to characterize these distributions. Extended, localized, and singular distributions are clearly distinguished only by the finite-size scaling behavior of $\alpha_{\rm min}$ and $f(\alpha_{\rm min})$. The multifractal spectrum of the singular distribution agrees well with that of a simple partitioning process.