  The spatial structure of a two-dimensional homogeneous mixture of fermionic
atoms in two hyperfine states is analyzed throughout the BEC-BCS crossover.
Within the BCS-Leggett mean-field model we consider three functions: the pair
wave function and the density-density correlation functions between atoms of
the same and of different hyperfine states. For the correlation functions we
derive analytical expressions which allow to unveil the rich spatial structure
of the superfluid. Mainly, we are able to study the large-distance behavior of
the three functions, which exhibits an exponential decay and a well-defined
oscillatory behavior. We report closed-form expressions for the correlation
lengths and mean pair radius. Differences and similarities emerge when
comparing with the three dimensional case. Particularly, we find an expression
for the large-distance correlation length, in terms of the chemical potential
and the gap, valid in two and three dimensions, but whose dependence on the
corresponding scattering lengths differ significantly.
