Segregation in the Falicov-Kimball model

The ground state of the one-dimensional Falicov-Kimball model (1969) is considered for large values of the interaction strength U. If the ion density rho i =p/q (p and q relatively prime) is rational and equals the electron density rho e the author proves that for U)U hom c (q) the ion configurations is the most homogeneous one. If 0( rho e ( rho i and U)U seg c ( rho e / rho i ) w prove that all the ions condense into one large cluster. This last result proves a recent conjecture known as the segregation principle.