  Limits of densities belonging to an exponential family appear in many
applications, {e.g.} Gibbs models in Statistical Physics, relaxed combinatorial
optimization, coding theory, critical likelihood computations, Bayes priors
with singular support, random generation of factorial designs. We discuss the
problem from the methodological point of view in the case of a finite state
space. We prove two characterizations of the limit distributions, both based on
a suitable description of the marginal polytope (convex hull of canonical
statistics' values). First, the set of limit densities is equal to the set of
conditional densities given a face of the marginal polytope. Second, in the
lattice case there exists a parametric presentation, in monomial form, of the
closure of the statistical model.
