  The ground state of an electron gas is characterized by the interparticle
spacing to the effective Bohr radius ratio r_s=a/a_B*. For polarized electrons
on a two dimensional square lattice with Coulomb repulsion, we study the
threshold value r_s* below which the lattice spacing s becomes a relevant scale
and r_s ceases to be the scaling parameter. For systems of small ratios s/a_B*,
s becomes only relevant at small r_s (large densities) where one has a quantum
fluid with a deformed Fermi surface. For systems of large s/a_B*, s plays also
a role at large r_s (small densities) where one has a Wigner solid, the lattice
limiting its harmonic vibrations. The thermodynamic limit of physical systems
of different a_B* is qualitatively discussed, before quantitatively studying
the lattice effects occurring at large r_s. Using a few particle system, we
compare exact numerical results obtained with a lattice and analytical
perturbative expansions obtained in the continuum limit. Three criteria giving
similar values for the lattice threshold $r_s^*$ are proposed. The first one is
a delocalization criterion in the Fock basis of lattice site orbitals. The
second one uses the persistent current which can depend on the interaction in a
lattice, while it becomes independent of the interaction in the continuum
limit. The third one takes into account the limit imposed by the lattice to the
harmonic vibrations of the electron solid.
