  The Falicov-Kimball model is a simple quantum lattice model that describes
light and heavy electrons interacting with an on-site repulsion; alternatively,
it is a model of itinerant electrons and fixed nuclei. It can be seen as a
simplification of the Hubbard model; by neglecting the kinetic (hopping) energy
of the spin up particles, one gets the Falicov-Kimball model.
  We show that away from half-filling, i.e. if the sum of the densities of both
kinds of particles differs from 1, the particles segregate at zero temperature
and for large enough repulsion. In the language of the Hubbard model, this
means creating two regions with a positive and a negative magnetization.
  Our key mathematical results are lower and upper bounds for the sum of the
lowest eigenvalues of the discrete Laplace operator in an arbitrary domain,
with Dirichlet boundary conditions. The lower bound consists of a bulk term,
independent of the shape of the domain, and of a term proportional to the
boundary. Therefore, one lowers the kinetic energy of the itinerant particles
by choosing a domain with a small boundary. For the Falicov-Kimball model, this
corresponds to having a single `compact' domain that has no heavy particles.
