  We study the large deviations behavior of systems that admit a certain form
of a product distribution, which is frequently encountered both in Physics and
in various information system models. First, to fix ideas, we demonstrate a
simple calculation of the large deviations rate function for a single
constraint (event). Under certain conditions, the behavior of this function is
shown to exhibit an analogue of Bose--Einstein condensation (BEC). More
interestingly, we also study the large deviations rate function associated with
two constraints (and the extension to any number of constraints is conceptually
straightforward). The phase diagram of this rate function is shown to exhibit
as many as seven phases, and it suggests a two--dimensional generalization of
the notion of BEC (or more generally, a multi--dimensional BEC). While the
results are illustrated for a simple model, the underlying principles are
actually rather general. We also discuss several applications and implications
pertaining to information system models.
