  Weak Wave Turbulence is a powerful theory to predict statistical observables
of diverse relevant physical phenomena, such as ocean waves,
magnetohydrodynamics and nonlinear optics. The theory is based upon an
asymptotic closure permitted in the limit of small nonlinearity. Here, we
explore the possible deviations from this mean-field framework, in terms of
anomalous scaling, focusing on the case of elastic plates. We establish the
picture of the possible behaviors at varying the extent of nonlinearity, and we
show that the mean-field theory is appropriate when all excited scales remain
dominated by linear dynamics. The other picture is non-trivial and our results
suggest that, when large scales contain much energy, the cascade sustains
extreme events at small scales and the system displays intermittency.
