  We investigate the stability with respect to phase separation or charge
density-wave formation of the two-dimensional Hubbard model for various values
of the local Coulomb repulsion and electron densities using Green-function
Monte Carlo techniques. The well known sign problem is particularly serious in
the relevant region of small hole doping. We show that the difference in
accuracy for different doping makes it very difficult to probe the phase
separation instability using only energy calculations, even in the
weak-coupling limit ($U=4t$) where reliable results are available. By contrast,
the knowledge of the charge correlation functions allows us to provide clear
evidence of a spatially homogeneous ground state up to $U=10t$.
