  We consider critical eigenstates in a two dimensional quasicrystal and their
evolution as a function of disorder. By exact diagonalization of finite size
systems we show that the evolution of properties of a typical wave-function is
non-monotonic. That is, disorder leads to states delocalizing, until a certain
crossover disorder strength is attained, after which they start to localize.
Although this non-monotonic behavior is only present in finite-size systems and
vanishes in the thermodynamic limit, the crossover disorder strength decreases
logarithmically slowly with system size, and is quite large even for very large
approximants. The non-monotonic evolution of spatial properties of eigenstates
can be observed in the anomalous dimensions of the wave-function amplitudes, in
their multifractal spectra, and in their dynamical properties. We compute the
two-point correlation functions of wave-function amplitudes and show that these
follow power laws in distance and energy, consistent with the idea that
wave-functions retain their multifractal structure on a scale which depends on
disorder strength. Dynamical properties are studied as a function of disorder.
We find that the diffusion exponents do not reflect the non-monotonic
wave-function evolution. Instead, they are essentially independent of disorder
until disorder increases beyond the crossover value, after which they decrease
rapidly, until the strong localization regime is reached. The differences
between our results and earlier studies on geometrically disordered
``phason-flip'' models lead us to propose that the two models are in different
universality classes. We conclude by discussing some implications of our
results for transport and a proposal for a Mott hopping mechanism between power
law localized wave-functions, in moderately disordered quasicrystals.
