  In this paper, we present a probabilistic study of rare phenomena of the
cubic nonlinear Schr\"odinger equation on the torus in a weakly nonlinear
setting. This equation has been used as a model to numerically study the
formation of rogue waves in deep sea. Our results are twofold: first, we
introduce a notion of criticality and prove a Large Deviations Principle (LDP)
for the subcritical and critical cases. Second, we study the most likely
initial conditions that lead to the formation of a rogue wave, from a
theoretical and numerical point of view. Finally, we propose several open
questions for future research.
