Large deviations principle for the cubic NLS equation with slowly decaying data

In this note, we prove a sharp large derivation principle (LDP) for the cubic nonlinear Schrödinger equation with Gaussian random initial data in Fourier Lebesgue spaces. As a consequence, we improve the exponential decay condition in [M.A. Garrido, R. Grande, K.M. Kurianski, G. Staffilani. Commun. Pure Appl. Math. 76 (2023), 4087–4136] to ${\mathrm{\ell }}^1$ decay.