  In our model, spinless fermions (or hardcore bosons) on a square lattice hop
to nearest neighbor sites, and also experience a hard-core repulsion at the
nearest neighbor separation. This is the simplest model of correlated electrons
and is more tractable for exact diagonalization than the Hubbard model. We
study systematically the dilute limit of this model by a combination of
analytical and several numerical approaches: the two-particle problem using
lattice Green functions and the t-matrix, the few-fermion problem using a
modified t-matrix (demonstrating that the interaction energy is well captured
by pairwise terms), and for bosons the fitting of the energy as a function of
density to Schick's analytical result for dilute hard disks. We present the
first systematic study for a strongly-interacting lattice model of the
t-matrix, which appears as the central object in older theories of the
existence of a two-dimensional Fermi liquid for dilute fermions with strong
interactions. For our model, we can (Lanczos) diagonalize the 7 by 7 system at
all fillings and the 20 by 20 system with four particles, thus going far beyond
previous diagonalization works on the Hubbard model.
