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It is well known that, if the initial conditions have sufficiently highenergy density, the dynamics of the classical Discrete Non-Linear Schr\"odingerEquation (DNLSE) on a lattice shows a form of breaking of ergodicity, with afinite fraction of the total charge accumulating on a few sites and residingthere for times that diverge quickly in the thermodynamic limit. In this paperwe show that this kind of localization can be attributed to some geometricproperties of the microcanonical potential energy surface, and that it can beassociated to a phase transition in the lowest eigenvalue of the Laplacian onsaid surface. We also show that the approximation of considering the phasespace motion on the potential energy surface only, with effective decoupling ofthe potential and kinetic partition functions, is justified in the largeconnectivity limit, or fully connected model. In this model we further observea synchronization transition, with a synchronized phase at low temperatures.

Localization in the Discrete Non-Linear Schr\"odinger Equation and geometric properties of the microcanonical surface