  We present a detailed account of a first-order localization transition in the
Discrete Nonlinear Schr\"odinger Equation, where the localized phase is
associated to the high energy region in parameter space. We show that, due to
ensemble inequivalence, this phase is thermodynamically stable only in the
microcanonical ensemble. In particular, we obtain an explicit expression of the
microcanonical entropy close to the transition line, located at infinite
temperature. This task is accomplished making use of large-deviation
techniques, that allow us to compute, in the limit of large system size, also
the subleading corrections to the microcanonical entropy. These subleading
terms are crucial ingredients to account for the first-order mechanism of the
transition, to compute its order parameter and to predict the existence of
negative temperatures in the localized phase. All of these features can be
viewed as signatures of a thermodynamic phase where the translational symmetry
is broken spontaneously due to a condensation mechanism yielding energy
fluctuations far away from equipartition: actually they prefer to participate
in the formation of nonlinear localized excitations (breathers), typically
containing a macroscopic fraction of the total energy.
