  It is well known that, if the initial conditions have sufficiently high
energy density, the dynamics of the classical Discrete Non-Linear Schr\"odinger
Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a
finite fraction of the total charge accumulating on a few sites and residing
there for times that diverge quickly in the thermodynamic limit. In this paper
we show that this kind of localization can be attributed to some geometric
properties of the microcanonical potential energy surface, and that it can be
associated to a phase transition in the lowest eigenvalue of the Laplacian on
said surface. We also show that the approximation of considering the phase
space motion on the potential energy surface only, with effective decoupling of
the potential and kinetic partition functions, is justified in the large
connectivity limit, or fully connected model. In this model we further observe
a synchronization transition, with a synchronized phase at low temperatures.
