  We consider a noninteracting disordered 1D quasicrystal in the weak disorder
regime. We show that the critical states of the pure model approach strong
localization in strikingly different ways, depending on their renormalization
properties. A finite size scaling analysis of the inverse participation ratios
of states (IPR) of the quasicrystal shows that they are described by several
kinds of scaling functions. While most states show a progressively increasing
IPR as a function of the scaling variable, other states exhibit a nonmonotonic
`re-entrant' behavior wherein the IPR first decreases, and passes through a
minimum, before increasing. This surprising behavior is explained in the
framework of perturbation renormalization group treatment, where wavefunctions
can be computed analytically as a function of the hopping amplitude ratio and
the disorder, however it is not specific to this model. Our results should help
to clarify results of recent studies of localization due to random and
quasiperiodic potentials.
