  We develop a finite temperature mean field theory in the path integral
picture for an extremely dilute system of interacting Fermions in a plane. In
the limit of short ranged interactions, the system is shown to undergo a phase
transition to a superconducting regime. Unlike the well known BCS transition
for metals, this phase transition is found to be very sensitive to the two
dimensional nature of the problem. A quantitative estimate of this sensitivity
is carried out by repeating the analysis at non-zero thickness. The validity of
the mean field results are also proved using renormalization group techniques.

Strongly Correlated Electrons;Superconductivity