  We present rigorous results for several variants of the Hubbard model in the
strong-coupling regime. We establish a mathematically controlled perturbation
expansion which shows how previously proposed effective interactions are, in
fact, leading-order terms of well defined (volume-independent) unitarily
equivalent interactions. In addition, in the very asymmetric (Falicov-Kimball)
regime, we are able to apply recently developed phase-diagram technology
(quantum Pirogov-Sinai theory) to conclude that the zero-temperature phase
diagrams obtained for the leading classical part remain valid, except for thin
excluded regions and small deformations, for the full-fledged quantum
interaction at zero or small temperature. Moreover, the phase diagram is stable
upon the addition of arbitrary, but sufficiently small, further quantum terms
that do not break the ground-states symmetries. This generalizes and unifies a
number of previous results on the subject; in particular published results on
the zero-temperature phase diagram of the Falikov-Kimball model (with and
without magnetic flux) are extended to small temperatures and/or small ionic
hopping. We give explicit expressions for the first few orders, in the hopping
amplitude, of these equivalent interactions, and we describe the resulting
phase diagram. Our approach, however, yields algorithms to compute arbitrary
high orders.
