  We have performed a Quantum Monte Carlo study of a two-dimensional bulk
sample of interacting 1/2-spin structureless fermions, a model of $^3$He
adsorbed on a variety of preplated graphite substrates. We have computed the
equation of state and the polarization energy using both the standard
fixed-node approximate technique and a formally exact methodology, relying on
bosonic imaginary-time correlation functions of operators suitably chosen in
order to extract fermionic energies. As the density increases, the fixed-node
approximation predicts a transition to an itinerant ferromagnetic fluid,
whereas the unbiased methodology indicates that the paramagnetic fluid is the
stable phase until crystallization takes place. We find that two-dimensional
$^3$He at zero temperature crystallizes from the paramagnetic fluid at a
density of 0.061 \AA$^{-2}$ with a narrow coexistence region of about 0.002
\AA$^{-2}$. Remarkably, the spin susceptibility turns out in very good
agreement with experiments.
