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

Nonmonotonic crossover and scaling behaviors in a disordered 1D quasicrystal