  The two dimensional crossover from independent particle towards collective
motion is studied using 2 spinless fermions interacting via a U/r Coulomb
repulsion in a LxL square lattice with periodic boundary conditions and nearest
neighbor hopping t. Three regimes characterize the ground state when U/t
increases. Firstly, when the fluctuation $\Delta r$ of the spacing r between
the two particles is larger than the lattice spacing, there is a scaling length
$Lo=\sqrt{8}\pi^2(t/U)$ such that the relative fluctuation $\Delta r/<r>$ is a
universal function of the dimensionless ratio L/Lo, up to finite size
corrections of order $L^{-2}$. L<Lo and L>Lo are respectively the limits of the
free particle Fermi motion and of the correlated motion of a Wigner molecule.
Secondly, when U/t exceeds a threshold U*(L)/t, $\Delta r$ becomes smalller
than the lattice spacing, giving rise to a correlated lattice regime where the
previous scaling breaks down and analytical expansions in powers of (t/U)
become valid. A weak random potential reduces the scaling length and favors the
correlated motion.
