  The Falicov-Kimball model consists of spinless electrons and classical
particles (ions) on a lattice. The electrons hop between nearest neighbor sites
while the ions do not. We consider the model with equal numbers of ions and
electrons and with a large on-site attractive force between ions and electrons.
For densities 1/4 and 1/5 the ion configuration in the ground state had been
proved to be periodic. We prove that for density 2/9 it is periodic as well.
However, for densities between 1/4 and 1/5 other than 2/9 we prove that the ion
configuration in the ground state is not periodic. Instead there is phase
separation. For densities in (1/5,2/9) the ground state ion configuration is a
mixture of the density 1/5 and 2/9 ground state ion configurations. For the
interval (2/9,1/4) it is a mixture of the density 2/9 and 1/4 ground states.
