  The motivation for this paper is the analysis of the fixed-density Ising
lattice gas in the presence of a gravitational field. This is a seen as a
particular instance of an Ising model with a slowly varying magnetic field in
the fixed magnetization ensemble. We first characterize the typical
magnetization profiles in the regime in which the contribution of the magnetic
field competes with the bulk energy term. We then discuss in more detail the
particular case of a gravitational field and the arising interfacial phenomena.
In particular, we identify the macroscopic profile and propose several
conjectures concerning the interface appearing in the phase coexistence regime.
The latter are supported by explicit computations in an effective model.
Finally, we state some conjectures concerning equilibrium crystal shapes in the
presence of a gravitational field, when the latter contributes to the energy
only to surface order.
