Rogue Waves and Large Deviations in the Nonlinear Schroedinger Equation with Random Initial Data Eric Vanden-Eijnden Courant Institute of Mathematical Sciences The problem of appearance of rogue waves is investigated within the context of the modified nonlinear Schroedinger (MNLS) equation with random initial data. Specifically, we identify the initial condition within a random set of given statistics that is most likely to create a large disturbance of the solution of MNLS within a certain time. This allows us to estimate the probability of appearance of such disturbances, and to estimate the tail of the distribution of the solution amplitude. To make contact with real observations, we assume that the initial condition is Gaussian distributed, with a spectrum taken from experimental data. The method proposed builds on results from large deviation theory and is transportable to other deterministic dynamical systems with random initial conditions.