  We study the superconducting proximity effect in inhomogeneous systems in
which a disordered or quasicrystalline normal-state wire is connected to a BCS
superconductor. We self-consistently compute the local superconducting order
parameters in the real space Bogoliubov-de Gennes framework for three cases,
namely, when states are i) extended, ii) localized or iii) critical. The
results show that the spatial decay of the superconducting order parameter as
one moves away from the normal-superconductor interface is power law in cases
i) and iii), stretched exponential in case ii). In the quasicrystalline case,
we observe self-similarity in the spatial modulation of the proximity-induced
superconducting order parameter. To characterize fluctuations, which are large
in these systems, we study the distribution functions of the order parameter at
the center of the normal region. These are Gaussian functions of the variable
(case i) or of its logarithm (cases ii and iii). We give arguments to explain
the characteristics of the distributions and their scaling with system size for
each of the three cases.

Superconductivity