  The presence of attractive interaction between fermions can lead to pairing
and superfluidity in an optical lattice. The temperature needed to observe
superfluidity is about a tenth of the tunneling energy in the optical lattice,
and currently beyond experimental reach. However, at strong coupling the
precursors to global superfluidity should be visible at achievable
temperatures, in terms of fluctuating domains with strong pairing correlations.
We explore this regime of the attractive two dimensional fermion Hubbard model,
in the presence of a confining potential, using a new Monte Carlo technique. We
capture the low temperature inhomogeneous superfluid state with its unusual
spectral signatures but mainly focus on the experimentally accessible
intermediate temperature state. In this regime, and for the trap center density
we consider, there is a large pairing amplitude at the center, spatially
correlated into domains extending over several lattice spacings. We map out the
thermal evolution of the local density, the double occupancy, the pairing
correlations, and the momentum distribution function across this phase
fluctuation window.
