  A general class of hopping models on a finite bipartite lattice is
considered, including the Hubbard model and the Falicov-Kimball model. For the
half-filled band, the single-particle density matrix $\uprho (x,y)$ in the
ground state and in the canonical and grand canonical ensembles is shown to be
constant on the diagonal $x=y$, and to vanish if $x \not=y$ and if $x$ and $y$
are on the same sublattice. For free electron hopping models, it is shown in
addition that there are no correlations between sites of the same sublattice in
any higher order density matrix. Physical implications are discussed.
