  We use exact diagonalization techniques to study the interplay between strong
correlations, superconductivity, and disorder in a model system. We study an
extension of the t-J model by adding an infinite-range d-wave superconductivity
inducing term and disorder. Our work shows that in the clean case the magnitude
of the order parameter is surprisingly small for low-hole filling, thus
implying that mean-field theories might be least accurate in that important
regime. We demonstrate that substantial disorder is required to destroy a
d-wave superconducting state for low-hole doping. We provide the first bias
free numerical results for the local density of states of a strongly correlated
d-wave superconducting model, relevant for STM measurements at various fillings
and disorders.
