Large Deviations for the Nonlinear Schrodinger Equation with Randomized Quasi-Periodic Initial Data in Higher Dimensions: Beyond the Critical Time Scale

We study the cubic weakly nonlinear Schr\"odinger equation with randomized spatially quasi-periodic initial data in higher dimensions. Under a polynomial decay assumption in Fourier space, we establish a {\em Large Deviations Principle} for rogue waves in the time regime $\mathcal O(\varepsilon^{-1-\eta})$ ($0 \le \eta < 1$), extending beyond the currently known critical time scale $\mathcal O(\varepsilon^{-1} |\log \varepsilon|)$ in the one-dimensional periodic setting \cite{GGKS23, FL25, LW25}. The proof proceeds in two main steps. We first characterize the distribution of the linear solution and establish the corresponding linear large deviations principle. The lower bound is obtained via pointwise estimates, while the upper bound follows from a combination of truncation and probabilistic arguments. We then perform a detailed combinatorial analysis of the Picard iteration, deriving an effective size for the Duhamel term and thus establishing the nonlinear large deviations principle.