  The two dimensional Hubbard model with a single spin-up electron interacting
with a finite density of spin-down electrons is studied using the quantum
  Monte Carlotechnique, a new conjugate gradient method for the evaluation of
the Edwards wavefunction ansatz, and the standard second order perturbation
theory. We performed simulations up to 242 sites at $U/t=4$ reaching the zero
temperature properties with no ``fermion sign problem'' and found a
surprisingly good accuracy of the Edwards wavefunction ansatz at low density or
low doping. The conjugate gradient method was then applied to system up to 1922
sites and infinite $U$ for the Edwards state. Fermi liquid theory seems to
remain stable in 2D for all cases studied with the exception of the half
filling case where a ``Luttinger like behavior'' survives in the Hubbard model
, yielding a vanishing quasiparticle weight in the thermodynamic limit.
